KOMISJA BUDOWY MASZYN PAN ODDZIAŁ W POZNANIU Vol. 27 nr 2 Archiwum Technologii Maszyn i Automatyzacji 27 MAŁGORZATA PONIATOWSKA * STUDY OF THE INFLUENCE OF SURFACE FORM ERRORS ON THE FITTING UNCERTAINTY IN COORDINATE MEASUREMENTS In coordinate measurements theoretical substitute features are determined using discrete data which are the coordinates of the measurement points. The fitting results have an uncertainty influenced by coordinate measuring machine, surface geometric errors and number and distribution of measurement points. The fitting uncertainty affects both the precision of the tolerance interval to be determined as well as the width of the conformity zone. The paper presents the investigations results of the influence of surface geometric errors on the fitting uncertainty of a least square circle. The investigations were performed on a surfaces superimposed by random form errors and systematic form errors for a various number of sample points uniformly distributed around the entire circle as well as on circular sectors defined by angles of different values. Key words: coordinate measurements, substitute feature, uncertainty, surface form errors. INTRODUCTION To ensure the production of high precision machine parts constitutes an essential element of modern manufacturing processes. Product quality inspections involve measurements of product dimensions and deviations followed by verifications of product specifications. At present coordinate measuring machines (CMMs) appear to be most technologically advanced and universal tools commonly used to perform all kinds of precision measurements. The versality allows CMMs to inspect a wide range of features and part types. Another advantage of CMMs is that the software designed to measure the dimensions of a specific object can be used a large number of times. As a result CMMs can be applied in industry for dimensional control of 3D objects of high complexity. * Dr inż. Chair of Materials Engineering and Mechanical Technology, Białystok Technical University.
82 M. Poniatowska In essence the coordinate measuring technique involves determining the location of discrete, finite number of sample points on the surface of the studied object. As a result, we obtain a set of data that are the coordinates of each and every measured point on the surface. The next step involves best fit i.e. determining a theoretical substitute feature. The measurement points collected from the true surface do not fit perfectly to the substitute feature. The best-fit results are affected by three major factors: the algorithm of the best fit, the number and distribution of measurement points and finally the effects of both CMM and surface geometry errors. Fitting uncertainty is important not only due to the dimensional analysis of the surface geometrical elements but also affects the accuracy of determining the datum features of the workpiece coordinate system [4]. A large number of theoretical investigations and experiments concerning the impact of CMM errors [3], best fit algorithms [] and sampling strategies [2] on the coordinate measurement results have been carried out. However not much has been published on the subject measurement uncertainties caused by surface geometry errors. Even if a perfect best-fit algorithm is used, surface geometry deviations as well as CMM errors will create some errors in the fitting process. Fitting uncertainty will decrease the tolerance interval determination accuracy and thus affect the conformity zone, as shown in Fig.. 3 4 2 4 3 U U Fig.. Effect of uncertainty on the measurement results assessment specification interval, 2 conformity interval, 3 nonconformity interval, 4 uncertainty intervals, U expanded uncer tainty Rys.. Konsekwencje niepewności podczas oceny wyników pomiaru; pole specyfikacji, 2 pole zgodności, 3 pola niezgodności, 4 przedziały niepewności, U niepewność rozszerzona Uncertainty assessment can be used to diagnose the fitting process in order to indicate the precision of determining of tolerance zones. Surface geometry errors of the analyzed object are generally much greater than the uncertainties of the measuring instrument. In this experiment, it has been assumed that the machine error is insignificant that it has no influence on the fitting results. The most commonly used surface geometry element for machine parts is the circle. It can be applied to describe a large number of internal and external cylindrical surfaces. The uncertainty of determining substitute features involves the uncertainty of defining the centre of the circle in X and Y di-
Study of the influence of surface form errors 83 rections as well as the uncertainty of the circle diameter and tolerance zone for both the diameter and form errors. In this paper the results of experimental investigations showing the influence of surface form errors on the uncertainty of determining both the centre X, Y coordinates and the diameter of the substitute least square circle (LSC) are presented. The investigated cylindrical surfaces showed similar form errors values, however they significantly differed in the very nature of the errors. Various numbers of sample points were used. The investigations were performed on CMM Mistral Standard 775 equipped with Renishaw TP2 probe with the stylus 2 mm length and a ball tip 2 mm in diameter. 2. SUBJECT, SCOPE AND METHODOLOGY OF EXPERIMENT 2.. Characteristics of form errors of the measured profiles The investigations were carried out on cylindrical surfaces at constant value of coordinate Z, thus in practice, reduced to 2D circles. The substitute circles were determined using the LSC method. In the first step a precise characteristics i.e. determination of the values and nature of geometry errors of the measured profiles was performed. The first cylinder of the cross-section roundness deviation Δ = μm had a profile presented in Fig. 2 (standard deviation of profile s =,52 μm) and the spectral structure of the profile shown in Fig. 3. There is no dominance of any harmonic component in the spectrum, which indicates a lack of the systematic form error. It can be assumed that the errors were of quasi-random distribution. 754 755 756 757 758 759 76 76 762 763 764 765 766 767 768 769 742 743 744 745 746 747 748 749 75 75 752 753 734 735 736 737 738 739 74 74 728 729 73 73 732 733 723 724 725 726 727 77 78 79 72 72 722 74 75 76 78 79 7 7 72 73 75 76 77 7 7 72 73 74 697 698 699 694 695 696 69 692 693 688 689 69 686 687 683 684 685 68 68 682 677 678 679 675 676 674 67 672 673 67 669 666 667 668 663 664 665 66 66 662 657 658 659 654 655 656 65 652 653 649 65 646 647 648 643 644 645 64 64 642 637 638 639 636 635 634 632 633 629 63 63 626 627 628 625 624 623 62 62 622 67 68 69 64 66 65 62 63 69 6 6 66 67 68 63 64 65 6 6 62 597 598 599 595 596 592 593 594 589 59 59 586 587 588 583 584 585 58 58 582 577 578 579 575 576 572 573 574 569 57 57 568 567 566 563 564 565 56 56 562 558 559 555 556 557 552 553 554 549 55 55 546 547 548 543 544 545 54 542 538 539 54 535 536 537 532 533 534 529 53 53 526 527 528 523 524 525 52 522 58 59 52 55 56 57 Δ=μm 3456789 3 2 6 5 4 92 8 7 2223 2 2526 24 2829 27 2 5-5 - -5-2 -25-3 52 53 54 33 3334 32 3637 35 39 38 4243 4 4 45 44 4748 46 55 49 53 52 5657 55 54 596 58 6263 6 6465 67 66 77 69 68 7374 72 76 75 798 78 77 8283 8 8485 87 86 99 89 88 93 92 96 95 94 99 98 97 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 32 33 34 35 36 37 38 39 4 4 44 43 42 45 46 47 48 49 5 5 52 53 56 55 54 57 58 59 6 6 62 63 64 67 66 65 68 69 7 7 72 73 74 75 76 77 78 79 8 8 82 83 84 85 86 87 88 9 89 92 9 93 94 95 96 99 97 2 98 2 24 23 22 25 28 26 29 27 2 2 23 22 24 25 26 29 27 22 28 22 222 225 223 226 224 228 229 227 232 235 23 234 23 238 237 233 24 24 236 239 242 245 243 246 244 248 249 247 25 253 25 254 252 256 257 255 259 26 258 262 263 26 265 266 264 267 27 268 27 269 275 274 273 272 278 28 277 28 276 284 283 279 282 285 287 286 29 288 29 289 293 294 295 292 299 3 3 296 297 298 34 35 36 32 33 32 3 3 37 38 39 33 36 34 37 35 39 32 38 32 324 322 325 323 327 328 329 326 333 334 335 33 33 332 339 34 336 337 338 344 345 346 34 342 343 35 35 352 349 348 347 356 357 353 354 355 36 362 363 36 359 358 367 368 369 364 365 366 373 374 375 372 37 37 378 379 38 376 377 387 388 389 384 385 386 38 382 383 395 396 397 394 393 39 39 392 4 42 43 398 399 4 4 4 42 47 48 49 44 45 46 45 46 47 43 44 424 425 426 42 422 423 48 49 42 432 433 434 43 43 427 428 429 44 442 443 438 439 44 435 436 437 455 456 457 452 453 454 449 45 45 448 447 444 445 446 469 47 47 467 468 464 465 466 463 462 46 458 459 46 495 496 497 492 493 494 49 49 489 486 487 488 485 484 48 482 483 478 479 48 475 476 477 472 473 474 59 5 5 58 57 56 54 55 5 52 53 498 499 5 Fig. 2. Roundness profile of the first circle Rys. 2. Odchyłki okrągłości pierwszego profilu
M. Poniatowska 84 harmonic amplitudes [μm],2,4,6,8 6 6 2 26 3 36 4 46 Fig. 3. Spectral structure of the first profile Rys. 3. Analiza widmowa pierwszego zarysu Δ=2μm -2-5 - -5 5 2 3456789 2 34 5 6 7 8 92 2 223 24 2526 27 2829 33 32 334 35 3637 38 39 4 4 4243 44 4546 4748 49 55 52 53 54 55 5657 58 596 6 6263 6465 66 67 68 69 77 72 7374 75 76 77 78 798 8 8283 8485 86 87 88 89 99 92 93 94 95 96 97 98 99 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 25 26 27 28 29 3 3 32 33 34 35 36 37 38 39 4 4 42 43 44 45 46 47 48 49 5 5 52 53 54 55 56 57 58 59 6 6 62 63 64 65 66 67 68 69 7 7 72 73 74 75 76 77 78 79 8 8 82 83 84 85 86 87 88 89 9 9 92 93 94 95 96 97 98 99 2 2 22 23 24 25 26 27 28 29 2 2 22 23 24 25 26 27 28 29 22 22 222 223 224 225 226 227 228 229 23 23 232 233 234 235 236 237 238 239 24 24 242 243 244 245 246 247 248 249 25 25 252 253 254 255 256 257 258 259 26 26 262 263 264 265 266 267 268 269 27 27 272 273 274 275 276 277 278 279 28 28 282 283 284 285 286 287 288 289 29 29 292 293 294 295 296 297 298 299 3 3 32 33 34 35 36 37 38 39 3 3 32 33 34 35 36 37 38 39 32 32 322 323 324 325 326 327 328 329 33 33 332 333 334 335 336 337 338 339 34 34 342 343 344 345 346 347 348 349 35 35 352 353 354 355 356 357 358 359 36 36 362 363 364 365 366 367 368 369 37 37 372 373 374 375 376 377 378 379 38 38 382 383 384 385 386 387 388 389 39 39 392 393 394 395 396 397 398 399 4 4 42 43 44 45 46 47 48 49 4 4 42 43 44 45 46 47 48 49 42 42 422 423 424 425 426 427 428 429 43 43 432 433 434 435 436 437 438 439 44 44 442 443 444 445 446 447 448 449 45 45 452 453 454 455 456 457 458 459 46 46 462 463 464 465 466 467 468 469 47 47 472 473 474 475 476 477 478 479 48 48 482 483 484 485 486 487 488 489 49 49 492 493 494 495 496 497 498 499 5 5 52 53 54 55 56 57 58 59 5 5 52 53 54 55 56 57 58 59 52 52 522 523 524 525 526 527 528 529 53 53 532 533 534 535 536 537 538 539 54 54 542 543 544 545 546 547 548 549 55 55 552 553 554 555 556 557 558 559 56 56 562 563 564 565 566 567 568 569 57 57 572 573 574 575 576 577 578 579 58 58 582 583 584 585 586 587 588 589 59 59 592 593 594 595 596 597 598 599 6 6 62 63 64 65 66 67 68 69 6 6 62 63 64 65 66 67 68 69 62 62 622 623 624 625 626 627 628 629 63 63 632 633 634 635 636 637 638 639 64 64 642 643 644 645 646 647 648 649 65 65 652 653 654 655 656 657 658 659 66 66 662 663 664 665 666 667 668 669 67 67 672 673 674 675 676 677 678 679 68 68 682 683 684 685 686 687 688 689 69 69 692 693 694 695 696 697 698 699 7 7 72 73 74 75 76 77 78 79 7 7 72 73 74 75 76 77 78 79 72 72 722 723 724 725 726 727 728 729 73 73 732 733 734 735 736 737 738 739 74 74 742 743 744 745 746 747 748 749 75 75 752 753 754 755 756 757 758 759 76 76 762 763 764 765 766 767 768 769 77 77 772 773 774 775 776 777 778 779 78 78 782 783 784 785 786 787 788 789 79 79 792 793 794 795 796 797 798 799 8 8 82 83 84 85 86 87 88 89 8 8 82 83 84 85 86 87 88 89 82 82 822 823 824 825 826 827 828 829 83 83 832 833 834 835 836 837 838 839 84 84 842 843 844 845 846 847 848 849 85 85 852 853 854 855 856 857 858 859 86 86 862 863 864 865 866 867 868 869 87 87 872 873 874 875 876 877 878 879 88 88 882 883 884 885 886 887 888 889 89 89 892 893 894 895 896 897 898 899 9 9 92 93 94 95 96 97 98 99 9 9 92 93 94 95 96 97 98 99 92 92 922 923 924 925 926 927 928 929 93 93 932 933 934 935 936 937 938 939 94 94 942 943 944 945 946 947 948 949 95 95 952 953 954 955 956 957 958 959 96 96 962 963 964 965 966 967 968 969 97 97 972 973 974 975 976 977 978 979 98 98 982 983 984 985 986 987 988 989 99 99 992 993 994 995 996 997 998 999 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 Fig. 4. Roundness profile of the second circle Rys. 4. Odchyłki okrągłości drugiego profilu harmonic amplitudes [μm],2,4,6,8,2,4 5 9 3 7 2 25 29 33 37 4 45 49 Fig. 5. Spectral structure of the second profile Rys. 5. Analiza widmowa drugiego zarysu
Study of the influence of surface form errors 85 The roundness deviation of the second cylinder was nearly the same i.e. Δ = = 2 μm (standard deviation of profile s =,38 μm) but the shape deviations were completely different in nature. The analysis of the diagrams presented in Figures 4 and 5 shows the presence of a three-lobed error. 2.2. Measurement strategy The origin of the part coordinate system was defined in the centre of the substitute circle consisting of 24 sample points, X,Y,Z axes aligned along the machine coordinate system. The investigations were concerned with the impact of the surface form errors on the fitting uncertainty. This is closely related to the number and distribution of sample points on the surface. For these reason the investigations were carried out for various (3, 4, 5, 6, 7, 8, 9,, 2, 5, 8, 24) number of sample points uniformly distributed on the circle. 36 repeats were made for each number of sample points. To eliminate the influence of points locations each set of points was rotated together with the part coordinate system about the origin by angle º with respect to the proceeding one. Making use of the strategy described above measurements were performed on the circles shown in section 2.. In engineering practice we often face a necessity of measuring on arc defined by a certain angle and radius e.g. a round edge. Owing to the above an attempt was made to find a solution to the following problem: what is the influence of the value of an arc angle on the uncertainty of determining of the arc s centre position and radius? To find a solution some investigations were made on a given angle. The measurements were focused on a circular sector whose subsequent angles were as follows: 45º, 6º, 75º, 9º, 2º, 8º. The number of sample points was constant and amounted to twelve. As in the case above the coordinate system was then rotated about the origin by angle 5º with respect to the machine coordinate system. 3. RESULTS AND DISCUSSION In this section the results of standard uncertainties evaluations in X and Y directions of the circle center positions and the standard uncertainties of the estimated diameters as well as the standard uncertainties of the arc radius of circular sectors. Fig. 6 shows the uncertainty of the X and Y centre position coordinates dependencies: ux, uy random profile and ux2, uy2 determined profile, versus number of sample points.
86 M. Poniatowska In the case of the first profile the uncertainties decrease starting from about μm for 3 sample points up to about,4 μm for 8 sample points and stay on the same level when the number of sample points is increased. standard uncertainties [μm],6,4,2,8,6,4,2 3 4 5 6 7 8 9 2345678922222324 number of sample points ux uy ux2 uy2 Fig. 6. Uncertainty of circle centre coordinates versus number of sample points Rys. 6. Zależność niepewności wyznaczania współrzędnych środka okręgu od liczby punktów pomiarowych In the case of three-lobed circle the standard uncertainties of the centre coordinates assume the highest values for 4 sample points i.e. about,5 μm and then decrease rapidly to assume a constant value below,4 μm when the sample size becomes larger then 9. Fig. 7 presents the curves of standard uncertainties variables for the estimation of circle diameters ud and ud2. 3 standard uncertainties [μm] 2,5 2,5,5 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 number of sample points ud ud2 Fig. 7. Uncertainty of circle diameter versus number of sample points Rys. 7. Zależność niepewności wyznaczania średnicy okręgu od liczby punktów pomiarowych
Study of the influence of surface form errors 87 The diagrams clearly illustrates the dependence between the uncertainty values and surface form errors for a small number of sample points, for 3 7 points ud2 value is twice higher than ud. From sample points the uncertainty values tend to be similar and assume the values below,5 μm. As a result it can be stated that above 9 sample points, disregarding the nature of surface geometry errors, the uncertainty assumes a constant value for both the circle centre coordinates and the diameter and is lower than the uncertainty of CMM. Figures 7 and 8 show the results of the uncertainties dependencies on the arc angle for 2 sample points. 2 5 5 standard uncertainties [μm] 45 6 75 9 5 2 35 5 65 8 angle of arc ux2 uy2 ux uy Fig. 8. Uncertainty of arc centre coordinates versus angle of arc Rys. 8. Zależność niepewności wyznaczania współrzędnych środka łuku od kąta łuku 25 standard uncertainties [μm] 2 5 5 45 6 75 9 5 2 35 5 65 8 angle of arc ur ur2 Fig. 9. Uncertainty of arc radius versus angle of arc Rys. 9. Zależność niepewności wyznaczania promienia od kąta łuku
88 M. Poniatowska The assessment uncertainty of the circle using a portion of the circle of threelobed error is four times higher than for a circle of random form error for small angles; Sx2 and Sy2 for 45º assume the values of several micrometers. This becomes even more evident on the diagram showing the uncertainties of radius determination for both Srand Sr2. All the uncertainties assume acceptable values only above 8º. 4. CONCLUSIONS In the paper the influence of surface geometric errors on the fitting uncertainties of substitute circles using LSM was investigated. The experimental investigations aimed at their practical application in engineering. The analysis of the results made it possible to state the following: surface form errors have a significant impact on fitting uncertainty. For a small number of sample points the fitting uncertainty of the substitute feature is several times higher in the case of profiles of systematic form errors than for random profiles, for profiles of systematic errors the uncertainty value changes in an abrupt way for a small number of measurement points. Fitting uncertainty, however, tends to decrease with an increase of the number of sample points in both cases, there is a certain measurement point number limit beyond which the fitting uncertainty value drops below the uncertainty of the CMM. This number is equal to nine for both types of profiles, i.e. systematic error and random profiles. Thus, it is possible to make a practical recommendation to use over nine sample points when performing measurements on the circular elements of typical machine parts, the uncertainty determined for the whole circle by using a randomly chosen arc section is considerably much higher than the uncertainty of the profile determined on the basis of the full circle (for the same number of measurement points). Here the critical value is the arc defined by 8 above which the uncertainty values approach the ones determined by the measurement of the point lying on the whole circumference of the circle. REFERENCES [] Chan F. M. M., King T. G., Stout K. J., The influence of sampling strategy on a circular feature in coordinate measurements, Measurement, 996, 9, 2, p. 73 8. [2] Dhanish P. B., Mathew J., Effects of CMM point coordinate uncertainty on uncertainties in determination of circular features, Measurement, 26, 39, p. 522 53.
Study of the influence of surface form errors 89 [3] Hong-Tzong Yau, Uncertainty analysis in geometric best fit, International Journal of Machine Tools and Manufacture, 998, 38, p. 323 34. [4] Qing Liu, Zhang C. C., Wang H. P. B., On the effects on CMM measurement error on form tolerance estimation, Measurement, 2, 3, p. 33 47. Praca wpłynęła do Redakcji 3.3.27 Recenzent: prof. dr inż. Jan Chajda BADANIE WPŁYWU BŁĘDÓW GEOMETRYCZNYCH POWIERZCHNI NA NIEPEWNOŚĆ DOPASOWANIA W POMIARACH WSPÓŁRZĘDNOŚCIOWYCH S t r e s z c z e n i e W pomiarach współrzędnościowych wyznacza się teoretyczne elementy zastępcze z danych dyskretnych, które są współrzędnymi lokalizowanych punktów pomiarowych. Wyniki dopasowania zawierają niepewność, na którą składają się wpływy błędów współrzędnościowej maszyny pomiarowej, błędów geometrycznych powierzchni, liczby i rozmieszczenia punktów pomiarowych. Niepewność dopasowania ma wpływ na dokładność wyznaczenia tolerancji oraz szerokość przedziału zgodności. W artykule przedstawiono wyniki badań wpływu błędów geometrycznych powierzchni na niepewność dopasowania okręgu średniego. Badania prowadzono na powierzchniach z błędami kształtu o charakterze losowym oraz o charakterze zdeterminowanym, dla różnej liczby punktów pomiarowych równomiernie rozłożonych na pełnym obwodzie, a także na wycinkach okręgów opisanych różnymi wartościami kątów. Słowa kluczowe: pomiary współrzędnościowe, element zastępczy, błędy kształtu, niepewność
9 M. Poniatowska