Vol. 27 nr 2 Archiwum Technologii Maszyn i Automatyzacji 2007
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 influ-enced 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 cir-cle. 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
1. INTRODUCTION
To ensure the production of high precision machine parts constitutes an es-sential element of modern manufacturing processes. Product quality inspections involve measurements of product dimensions and deviations followed by verifi-cations of product specifiverifi-cations.
At present coordinate measuring machines (CMMs) appear to be most tech-nologically 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 de-signed to measure the dimensions of a specific object can be used a large num-ber of times. As a result CMMs can be applied in industry for dimensional con-trol of 3D objects of high complexity.
*
Dr inż. – Chair of Materials Engineering and Mechanical Technology, Białystok Technical University.
In essence the coordinate measuring technique involves determining the loca-tion 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. deter-mining 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 [1] 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 uncer-tainty will decrease the tolerance interval determination accuracy and thus affect the conformity zone, as shown in Fig. 1.
Fig. 1. Effect of uncertainty on the measurement results assessment 1 – specification interval, 2 – conformity interval, 3 – nonconformity interval, 4 – uncertainty intervals, U – expanded uncer tainty
Rys. 1. Konsekwencje niepewności podczas oceny wyników pomiaru; 1 – 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 ma-chine parts is the circle. It can be applied to describe a large number of internal and external cylindrical surfaces. The uncertainty of determining substitute fea-tures involves the uncertainty of defining the centre of the circle in X and Y
di-1
4 4
3 2 3
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 pre-sented. 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 070705 equipped with Renishaw TP20 probe with the stylus 20 mm length and a ball tip 2 mm in diameter.
2. SUBJECT, SCOPE AND METHODOLOGY OF EXPERIMENT
2.1. 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 = 11 μm had a profile presented in Fig. 2 (standard deviation of profile s = 1,52 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.
Δ=11μm -30 -25 -20 -15 -10 -5 0 5 10 1 2 3 4 56789101112131415161718192021222524232627282930333231343536373839424041434445464748495051525354555659585760616263646566676869707172737675747778798081828483858687888990959691949392979899100101102103104105106107110109108111112113114115116117118119120121122123124127126125128129130131132133136135134137138139140141142143144147146145149150148151152153158157156155154159160161162163164169170165168167166171172173174175176177178179180181186187182185183184188189190191192193194195199200201198196197207210209208206205204203202211212213214215220221216219217218222223224225226227228229277274275276283278279280281282273284241230231232233234235236237238239240269267268242270271272265266263264253244245246247248249250251254243255256257258259260261262252 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769
Fig. 2. Roundness profile of the first circle Rys. 2.Odchyłki okrągłości pierwszego profilu
harmonic amplitudes [μm] 0 0,2 0,4 0,6 0,8 1 1 6 11 16 21 26 31 36 41 46
Fig. 3. Spectral structure of the first profile Rys. 3. Analiza widmowa pierwszego zarysu
Δ=12μm -20 -15 -10 -5 0 5 1 2 3 4 56789101112131415161918172021222324252826273029313233343536414237403938434445474649504851525354555659575860616263647069676866657172737475767978778081828384858687908988919293949596999897100101102103104105106107110109108113111112114115116117118119120121123124122125126127128129130133131132135136134137138139140141144142143145146147148149150153152151154155156157158159160161164163162165166167168169170171172173 174 175180181176179178177182183184185186187192191190188189193194195196197198199200201204202203206207208209210205214215213212211223228227226225224217222221220219218216229 230 231 232 233 234 235 236 237 238 239 240 241258271270269268267266265264263262261260259257242256255254253252251250249248247246245244243272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784785786787788789790791792793794795796797798 799 800 801802803 804806807808809810805811812813814815820821817819818816822823824825826827828829830831832833834835836837838839840841847846845844843842848849850851852853854855856857858 859 860 861863864862865866867868869872870871873874875876877878881880879882883884885886887892891890889888893894895896897898901900899902903904905906909908907910911912913914915916917918921920919922923924925926927928929932931930933934935936937938939940941942943944945946947948949952951950953954955956957958959960963962961964965966967968969975974972973971970976977978979980983981982984985986987988989992991990993994995100099699710021003100199999810041005100610071008100910121011101010131014101510161017101810191020102110221023 1024
Fig. 4. Roundness profile of the second circle Rys. 4. Odchyłki okrągłości drugiego profilu
harmonic amplitudes [μm] 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1 5 9 13 17 21 25 29 33 37 41 45 49
Fig. 5. Spectral structure of the second profile Rys. 5. Analiza widmowa drugiego zarysu
The roundness deviation of the second cylinder was nearly the same i.e. Δ = = 12 μm (standard deviation of profile s = 1,38 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 sub-stitute circle consisting of 1024 sample points, X,Y,Z axes aligned along the ma-chine 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, 10, 12, 15, 18, 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 10º with respect to the proceeding one.
Making use of the strategy described above measurements were performed on the circles shown in section 2.1.
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 subse-quent angles were as follows: 45º, 60º, 75º, 90º, 120º, 180º. The number of sam-ple points was constant and amounted to twelve. As in the case above the coor-dinate system was then rotated about the origin by angle 15º with respect to the machine coordinate system.
3. RESULTS AND DISCUSSION
In this section the results of standard uncertainties evaluations in X and Y direc-tions of the circle center posidirec-tions 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 de-pendencies: ux1, uy1 – random profile and ux2, uy2 – determined profile, versus number of sample points.
In the case of the first profile the uncertainties decrease starting from about 1 μm for 3 sample points up to about 0,4 μm for 8 sample points and stay on the same level when the number of sample points is increased.
standard uncertainties [μm] 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 number of sample points
ux1 uy1 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 coor-dinates assume the highest values for 4 sample points i.e. about 1,5 μm and then decrease rapidly to assume a constant value below 0,4 μm when the sample size becomes larger then 9.
Fig. 7 presents the curves of standard uncertainties variables for the estima-tion of circle diameters ud1 and ud2.
standard uncertainties [μm] 0 0,5 1 1,5 2 2,5 3 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 number of sample points
ud1 ud2
Fig. 7. Uncertainty of circle diameter versus number of sample points
The diagrams clearly illustrates the dependence between the uncertainty val-ues and surface form errors for a small number of sample points, for 3–7 points ud2 value is twice higher than ud1. From 10 sample points the uncertainty val-ues tend to be similar and assume the valval-ues below 0,5 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 12 sample points.
standard uncertainties [μm] 0 5 10 15 20 45 60 75 90 105 120 135 150 165 180 angle of arc
ux2 uy2 ux1 uy1
Fig. 8. Uncertainty of arc centre coordinates versus angle of arc
Rys. 8. Zależność niepewności wyznaczania współrzędnych środka łuku od kąta łuku
standard uncertainties [μm] 0 5 10 15 20 25 45 60 75 90 105 120 135 150 165 180 angle of arc ur1 ur2
Fig. 9. Uncertainty of arc radius versus angle of arc Rys. 9. Zależność niepewności wyznaczania promienia od kąta łuku
The assessment uncertainty of the circle using a portion of the circle of three-lobed 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 be-comes even more evident on the diagram showing the uncertainties of radius determination for both Sr1and Sr2. All the uncertainties assume acceptable val-ues only above 180º.
4. CONCLUSIONS
In the paper the influence of surface geometric errors on the fitting uncertain-ties of substitute circles using LSM was investigated. The experimental investi-gations 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 fit-ting 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 ma-chine parts,
− the uncertainty determined for the whole circle by using a randomly cho-sen 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 180° above which the uncer-tainty values approach the ones determined by the measurement of the point lying on the whole circumference of the circle.
REFERENCES
[1] Chan F. M. M., King T. G., Stout K. J., The influence of sampling strategy on a circular feature in coordinate measurements, Measurement, 1996, 19, 2, p. 73–81.
[2] Dhanish P. B., Mathew J., Effects of CMM point coordinate uncertainty on uncertainties in determination of circular features, Measurement, 2006, 39, p. 522–531.
[3] Hong-Tzong Yau, Uncertainty analysis in geometric best fit, International Journal of Ma-chine Tools and Manufacture, 1998, 38, p. 1323–1341.
[4] Qing Liu, Zhang C. C., Wang H. P. B., On the effects on CMM measurement error on form tolerance estimation, Measurement, 2001, 30, p. 33–47.
Praca wpłynęła do Redakcji 30.03.2007 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 dopasowa-nia zawierają niepewność, na którą składają się wpływy błędów współrzędnościowej maszyny pomiarowej, błędów geometrycznych powierzchni, liczby i rozmieszczenia punktów pomiaro-wych. 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 powierzch-niach z błędami kształtu o charakterze losowym oraz o charakterze zdeterminowanym, dla różnej liczby punktów pomiarowych równomiernie rozłożonych na pełnym obwodzie, a także na wycin-kach okręgów opisanych różnymi wartościami kątów.