5Th International DAAAM Baltic Conference s1

5th International DAAAM Baltic Conference

“INDUSTRIAL ENGINEERING – ADDING INNOVATION CAPACITY OF LABOR

FORCE AND ENTEREPRENEURS”

20-22 April 2006, Tallinn, Estonia

MEASUREMENT UNCERTAINTY OF SURFACE CONTOUR WITH

COMPLICATED FORM

Abiline, I., Laaneots, R., Nanits, M. & Riim, J.

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Abstract: Modern manufacture per-forms also to the accuracy of surfaces of the details with complicated form quite high demands. We can estimate the conformity of the produced details form to the demands of technical specification only by measurement. It is possible in Tallinn University of Technology laboratory of metrology to measure the surface contour of details with complicated form using the inductive surface roughness and form measurement instrument “Perthometer Concept”. Aforementioned measuring instrument has previously calibrated, but to confirm the reliability of the measurement results characterizing the surface contour we carried out in the laboratory of metrology the special investigation. In the measurement of the surface contour has composed the measurement model and has experi-mentally determined the values of the input quantities and their distribution. As a result of the research work in TTU laboratory of metrology we can charac-terize the reliability of the measurement results of the surface contour measurement with the expanded uncertainty.

Key words: surface form, measurement instrument, result of measurement, uncertainty of measurement.

1. INTRODUCTION

According to the reference document [1], the first step to determine the uncertainty of a measurement is to calculate the model function f that shows the relationship bet-

ween the input quantities (X1, X2, …,XN)

and the quantity to be measured.

(1)

The model function f represents the pro-cedure of the measurement and the method of evaluation. It describes how values of the output quantity Y are obtained from values of the input quantities Xi. In most cases it will be an analytical expression, but it may also be a group of such expressions which include corrections and correction factors for systematic effects, thereby leading to a more complicated relationship that is not written down as one function explicitly. Further, f may be determined numerically, or it may be a combination of all of these.

An estimate of the measurand Y, the output estimate denoted by y, is obtained from equation (1) using input estimates xi for the values of the input quantities Xi

(2)

It is understood that the input values are best estimates that have been corrected for all effects significant for the model. If not, the necessary corrections have been introduced as separate input quantities.

The lack of knowledge causes that the estimations of the input quantities are not exact, giving place to the concept of uncertainty, characterized by the standard deviation of the output quantity Y. The calculation of the output quantity is carried out applying the law of propagation of variances to (1), being its equation, if the

input quantities are independent:

(3)

It is necessary, therefore, to know the standard deviations, called standard uncertainties, of each one of the input quantities (u(Xi)). Depending on how the standard uncertainty is estimation, the set of input quantities may be grouped into two categories [1, 2]:

- Type A evaluation.

The standard uncertainty of input quantities is evaluated by statistical analysis of a series of observations.

- Type B evaluation.

The standard uncertainty of input quantities is evaluated by mean of tools different from the statistical analysis of a series of observations. In this case the information can come from the following sources: calibration certificates, handbooks, producer’s specifications and hypothesis on the function of density of the input quantity. Calculated the standard uncertainty of the output quantity with the law of propagation of variances, the expanded uncertainty of measurement U is obtained multiplying the standard uncertainty by a coverage factor k.

(4)

The value of k depends on the probability distribution of the output quantity y and on the level of confidence. The assigned expanded uncertainty corresponds to a coverage probability of approximately 95%. To locate the value of the measurement inside the interval it was defined by ±U. In most of the calibrations the output distribution can approach to a normal distribution, where k = 2.

2. MEASUREMENT METHOD OF THE CONTOUR AND EQUIPMENT

We measured the abovementioned surface
contours with complicated form using the surface texture measuring system “Pertho-meter Concept” made by company MAHR [3].

Fig.1 Measurement schema. 1 – measuring object; 2 – stylus; 3 – tracing arm; 4 – drive unit; 5 – measuring direction; 6 – ca-

librated support.

Fig.2 Tracing of the measuring object.

Tallinn University of Technology labora-tory of metrology owns this system about one year. Perthometer Concept is a modu-lar computer-controlled station for measu-ring and analysing roughness, contour and topography. The Perthometer Concept software runs under the worldwide Windows user interface. Operation is therefore quickly learned, easy to understand, and compatible with other Windows applications. We used for our research the PCV 200 contour drive unit with exchangeable tracing arm. The high-precision PCV 200 contour drive unit is a long-distance instrument for the assessment of radii, distances, angles and straightness deviations. The smooth traverse and the computer-assisted error correction guarantee reproducible measurements with utmost vertical and horizontal resolution in a measuring field of 200 mm x 50 mm. The PCV 200 contour drive unit allows automatic lowering and lifting of the tracing arm with programmable speed and quick positioning. Measuring force can be adjusted from 2 mN to 120 mN. Rigid design and unique material provide highly dynamic construction. Drive unit has programmable measuring routines inclu-ding lowering, lifting and positioning of the tracing arm and selectable measuring speeds.

Fig. 3 Surface texture measurement system “Perthometer Concept”

3. MEASUREMENT MODEL

Issued from the equation (2) we can express the measurement model as follows:

(5)

where:

x – measurement value

where

Now we can express the measurement model by the equation:

(6)

where: – correction from the

measuring instrument

– correction from the stylus

radius

– measurement force correcti-

– surface curvature correction

– surface concavity correction

– correction from the surface

angle

Combined standard uncertainty is deter-mined as:

(7)

4. RESEARCH RESULTS

We determined the input quantities standard uncertainties from the different sources.

Indication (contour) x

Indication in our case is a contour we see in the screen of the computer (see Fig. 3) Standard uncertainty of the indication can be determined according to the printer resolution. Our printer resolution

= 0.6 µm

Measuring instrument correction

Mentioned correction we didn’t found in the calibration certificate, but there was mentioned, that indication can change in the limits of

Fig. 4 Measured contour

Stylus radius correction

Research results showed us, that stylus radius correction didn’t effected remark-able to the contour measurements. So we can take:

and

Measuring force correction

Measuring force correction and its standard uncertainty will be calculated as follows. From the Hertz formula will be calculated the elastic deformation. We observed the worst situation: sphere-sphere. Correction value will be taken equal to zero and its standard uncertainty will be calculated by the equation:

where

Surface complexity correction ,

The correction due to the surface curvature and concavity will be taken equal to zero.

and

The standard uncertainty of these corrections can be calculated from the equation:

where and have found experi-mentally.

Correction of the surface angle

And its standard uncertainty can be calculated from the equation:

where has determined during the research experimentally using the angle standards.

Beforementioned quantities and their values have performed in the following table.

Quantity
Xi / Estimate
xi / Standard
uncertainty
u(xi), µm / Disper-
sion
u2(xi)
x / contour / 0.6 / 0.36
/ 0 / 0.3 / 0.09
/ 0 / 0 / 0
/ 0 / 0.3 / 0.09
/ 0 / 0.9 / 0.81
/ 0 / 0.9 / 0.81
/ 0 / 1.7 / 2.89
∑ / 4.7 / 5.05

Table 1. Research results

From the equation (7) we can calculate:

Now we can give the expanded uncertainty according to the equation (4):

5. CONCLUSIONS

As a result of current research work it is possible to give an estimation to the surface elements we got in printout after measuring the complicated surface contour. Also we can estimate the variation range and analyse the limits, where the numerical values of the surface contour can change. Finally we can evaluate the quality of the measuring values.

6. REFERENCES

1. Guide to the Expression of Uncertainty

in Measurement, first edition, 1993,

corrected and reprinted 1995, Inter-

national Organization for Standardiza-

tion (Geneva, Switzerland).

2. International Vocabulary of Basic and

General Terms in Metrology, second

edition, 1993, International Organiza-

tion for Standardization (Geneva,

Switzerland).

3. Perthometer Concept. Operating

Instructions. MAHR GmbH, Germany

7. CORRESPONDING AUTHOR

Indrek Abiline, M.Sc.

Department of Metrology,

Institute of Mechatronics,

Tallinn University of Technology

Ehitajate tee 5, 19086 Tallinn, Estonia

Phone: Int +372 6545181

Fax: Int +372 6770304

E-mail:

8. Acknowledgement

The work is carried out with the support of the Estonian Ministry of Education Science Grant No 14250503 and Estonian Science Grant No 6172.

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