XXI IMEKO World Congress Measurement in Research and Industry August 30 eptember 4, 05, Prague, Czech Republic UNCERTAINTY EVALUATION OF INUOIDAL FORCE MEAUREMENT Christian chlegel, Gabriela Kiekenap,Rol Kumme Physikalisch-Technische Bundesanstalt, Braunschweig, Germany, Christian.chlegel@ptb.de, Gabriela.Kiekenap@ptb.de, Rol.Kumme@ptb.de Abstract Periodical orce measurement is used or the primary calibration o orce transducers. This abstract describes models or an uncertainty determination o the sensitivity, which is the main measurand. Inluences like the mechanical coupling o test masses as well as their rocking during a measurement are the dominating contributions to the uncertainty o measurement. The achievable uncertainties will be demonstrated on the basis o intercomparison periodical orce measurements between CEM and PTB as well as the comparison between standard GUM and Monte Carlo based uncertainty evaluation Keywords: dynamic orce measurement, acceleration measurement, uncertainty evaluation, key comparison, Monte Carlo method.. INTRODUCTION TO THE METHOD Periodical orce measurement is one o the primary methods or the realization o dynamic orce, whereby the orce is determined according to Newtons Law, mass times acceleration. The traceability is thereby realized by an acceleration measurement based on a laser vibrometer and a mass measurement. The laser vibrometer is ussually operated with a laser wave length o 63.8 nm. The vibrometer used here at PTB was calibrated against the national acceleration standard or vibrometer calibration. The masses are determined with a balance, calibrated with a set o reerence masses calibrated against the national mass standard. The measurand which is determined in a periodical orce calibration is the dynamic sensitivity. The sensitivity is thereby the ratio o the output signal o the orce transducer and the dynamic orce. To obtain the pure calibration o the orce sensor, the requency response o the conditioning ampliier has to be unolded rom the electrical signal. For the dynamic calibration o bridge ampliiers a special bridge calibration standard is used, which was calibrated against the national voltage standard. In the case o piezoelectric transducers the charge ampliiers are calibrated against a reerence capacity. It should be noted that all calibrations have to be done or the requency range used in the orce measurement. The sinusoidal calibration is perormed in such a way that all requencies which are used or a calibration are measured separately. Thereby exactly a certain number o sinusoidal periods will be used to avoid ilter windows or the determination o the amplitudes in the FFT analysis o the measured time signals. The excitation signal is generated via an arbitrary waveorm generator module, which is then directly ed in the power ampliier o an electrodynamic shaker, where the orce transducer is mounted. The transducer itsel is equipped with an additional loading mass to generate, together with the adjusted acceleration, a certain dynamic orce. More details o the method and signal analysis can be ound in [].. MEAUREMENT MODEL For the determination o the uncertainty o the sensitivity the ollowing model is used: U ku ( M + m ) a k i a K Thereby is the requency-dependent sensitivity given in mv/v/n in case o a strain gauge orce transducer, and in pc/n in case o a piezoelectric transducer. The output signal o the conditioning ampliier is U, and k U is the calibration actor o the ampliier at the certain excitation measuring requency. The denominator contains M the mass o the orce initiating test mass or load mass, and m i is the head mass or internal mass o the orce transducer. The acceleration measured on the surace o the loading mass is a, and k a is the calibration actor or the acceleration o the vibrometer at a certain requency. Finally K corr is a correction actor o the acceleration, taking into account the vertical acceleration gradient o the loading mass. It should be noted that this model is only valid i the loading mass is directly screwed onto the orce transducer. As mentioned above, i one is using an adapter to dock the loading mass on the transducer, also the acceleration o the adapter has to be measured, due to the relative motion o the adapter in respect to the loading mass. In this case the ollowing model has to be applied: ( m a i a corr U ku + m ) a k + M a k a K corr () ()
Thereby a is the acceleration measured on the adapter and a the acceleration measured on the loading mass, whereby k a and k a are the associated calibration actors. The mass o the adapter is given by m a. It should be noted that the accelerations as well as the dedicated orce signal will be obtained by averaging several scan points, distributed over the surace o the whole mass block or adapter. To distinguish the inluence o the adapter rom the test mass, it is useul to separate the sensitivity into two contributions. U ( ma + mi ) a U M a (3) Thereby is the sensitivity which is measured on the adapter and the sensitivity determined on the mass block. As seen later on, the behaviour o these two values will be quite dierent. Both sensitivities can then be combined to the inal sensitivity or the orce transducer: ; ( ma + mi ) a M a + U U 443 4 3 For more clearness the calibration constants and the correction actor or the vertical acceleration gradient are not written in either part o equations (4), to both parts o equation (3), but have to be considered in the analysis. As seen rom the structure o the inal sensitivity,, is just a parallel connection o the individual sensitivities, and. On the other hand the sensitivity can also be expressed in terms o the parameters, stiness k and damping actor b o the orce transducer. In the simplest case the transducer can be approximated by two masses (bottom and top mass), which are connected by a spring. In this case the sensitivity is given by: 0 (5) b + ω k Thereby 0 is the statically determined sensitivity. For practical use equation (5) can be developed in a Taylor series: 3 4 5 4 6 0 λω + λ ω λ ω +... (6) 8 6 Thereby λb / k, the ratio o stiness to damping actor. As seen later on, this kind o polynomial can be used to it the measured data o the sensitivity. + (4) 3. MEAURED ENITIVITY To illustrate the dierent contributions according to equations (3), an example o measurement is given in Fig.. The measurement was done with a strain gauge transducer with a nominal orce o. kn. The loading mass o 6 kg was mounted onto the transducer with a mechanical adapter (mass approx. kg). The adapter is based on two clamping bushes which clamp the mass block to the transducer, see Fig. The acceleration was measured with the aid o a scanning vibrometer. The scanning vibrometer is able to measure, on a certain given area, many acceleration points in a sequential way. Usually a grid o points is deined on the surace o the object to be measured. In our case, this was one grid on the ring surace o the mass block and another one on the circular surace o the inner shat o the adapter. Thereby the mass block grid had 8 points and the adapter grid 9 points. The measuring procedure is the ollowing; in a irst scanning run all points are measured one ater the other. Aterwards the sotware will check i really all points are valid, i not, invalid points will be measured again. A invalidated measurement may or instance be an noisy signal, where the sotware cannot really determine the acceleration amplitude. The whole procedure will be repeated three times, and as the inal acceleration value the mean value is taken. Fig.. Mechanism to clamp a mass block with a mechanical adapter to a orce transducer. One clamping bush clamps the mass block, the other one clamps both to a cylindrical shat, which is screwed onto the orce transducer. In Fig. one can see that the behaviour o the two sensitivities is quite dierent. The sensitivity on the adapter is more or less constant with increasing requency; in contrast the sensitivity measured on the mass block decreases as a unction o requency. The combined sensitivity,, is at lower requencies, which is in accordance with the statically determined sensitivity. Figure shows also that in the case where an adapter is used, it is necessary to take into account the adapter behaviour. I one only measured on the mass block, the sensitivity would be overestimated.
Fig.. The sensitivity o a strain gauge transducer which was equipped with a 7 kg loading mass is shown. The sensitivity was devided to the two contributions according equation (3). In addition the statically measured sensitivity is shown as a constant line. In the ramework o the EMRP Programme Dynamic Measurement o Mechanical Quantities [] a comparison o sinusoidal orce measurement was perormed between CEM and PTB; thereby the measurement setups were quite similar. The main dierence was that CEM used a single point vibrometer. Nevertheless, also at CEM there were several acceleration measurements on the surace o the mass block by moving the vibrometer. In Fig. 3 the result comparing CEM and PTB regarding the sensitivity measurement is shown or the case o the transducer described above. For the coupling o the 6 kg test mass the same adapter was used. It should be noted that the analysis o the data was undertaken with dierent data acquisition systems. Fig.4. Relative deviation o the sensitivity in percent between CEM and PTB. more clearly, i the relative deviation o the sensitivity between both laboratories is compared. According to Fig. 4 one can summarize that or the investigated case the deviation o the combined sensitivity is % up to requencies o.5 khz. The origin o the bigger deviations can be seen in mechanical inluences between the orce transducer on one side and the adapter and the test mass on the other side. As shown in [3] there is a greater probability or the rocking o the test mass against the transducer body. This rocking may be included in the standard deviation o the acceleration measurement i the surace o the mass body is scanned with several acceleration points. As also shown in [3] the direction o rocking may change with requency and can be also quite asymmetric. Because o the act that the CEM was only able to measure the acceleration on a ew selected points (4-5), the eect o rocking cannot be detected careully. From this point o view it is not surprising that bigger deviations occur at higher requencies. 4. UNCERTAINTIE BAED ON TANDARD GUM Fig.3. Comparison between CEM and PTB o the sensitivity rom the transducer described already in Fig.. According to Fig. 3 the behaviour o the sensitivity curves is quite similar or CEM and PTB. At requencies above khz a bigger deviation can be seen. This can be shown even Figure 5 shows the uncertainties o the two sensitivities and, the uncertainty o the orce value as well as the uncertainty o the combined sensitivity. The main contribution to the uncertainty arises rom the averaged accelerations. The mass value o the test mass and the adapter was determined with a relative uncertainty o 5 0-3 %, whereby the internal mass was determined with %. As shown in [] the relative uncertainty connected with the calibration o the requency response o the bridge ampliier used can be given with 0.03 %. The vibrometer calibration contributes 0. % to the measured acceleration values. The propagation o uncertainty was derived according to the standard GUM using the model in accordance with equations -4. As shown in Fig. 5 the main contribution to the whole sensitivity originate rom the uncertainty, connected with the mass block. This is consistent with the act that the mass block is more inluenced by rocking
Fig. 6. ensitivity data together with a polynomial itting curve according equation (7) in red. In addition the conidence band o the it unction or 95% conidence (green) is shown. The solid blue line is a polynomial it through the error bars o the data points and the dashed blue line the lower and upper conidence level (95%) o this it. band (green). The same it can be applied or the error bars o the data (solid blue) which also gives a conidence (95%) band (dashed blue). Fig. 5. Uncertainty distributions o the main measurement components which are the contributions and, the orce signal, as well as the inal uncertainty o the combined sensitivity. motions compared with the adapter, which is more rigid and more compact in the middle o the excitation axis. The uncertainty contribution o the orce signal is small compared with the acceleration uncertainty. 4. APPROXIMATION OF THE ENITIVITY Calibration data are more comortable to handle i they can be described by an analytical unction. Oten the calibration or a sensor has to be deposited in the sotware o a machine or incorporated in measuring programs. For that reason, a polynomial it o the calibration data may be provided in accordance with equation (6). In Fig. 6 the ollowing polynomial was itted to the data: 4 6 ( a x + a x a ) y a (7) 0 3x Thereby x is the requency and y the sensitivity. In Fig. 6 the it (red) is shown together with his conidence (95%) 5. UNCERTAINTY BAED ON MC-IMULATION According the GUM upplement the output uncertainty o a model can also be obtained by the propagation o the Probability Density Functions (PDF) o the input quantities. This approach has the advantage, that the uncertainty o the output quantity can derived in a more natural way rom the standard deviation o the output PDF. The k-expansion actor is not needed because a coverage interval can be obtained, with the ad o the output PDF. The amount o output quantities which all in a certain probability interval can be counted. In the case o a non-symmetric output PDF the coverage interval is normally also non symmetric and is more realistic then the error region obtained with the standard GUM approach. For the model shown in eq. dierent kind o input PDFs were used. In the cases, where repeated measurements were perormed a t-distribution was applied. The reason is, that in the case o a inite number o measurements the estimate o the standard deviation can be more realistic described with a t-distribution. As shown above, the acceleration measured on the mass block were obtained by averaging 8 scanning points, the acceleration on the adapter by 9 points and the orce signal by the sum o both, 37 measurements. In addition the uncertainty connected with the calibration o the vibrometer and the bridge ampliiers were taken into account with a rectangular distribution. The other input values, the masses o the mass block, the adapter and the internal mass o the orce transducer were also modelled with a rectangular distribution.
Fig. 8. Output PDFs o the sensitivity obtained at dierent requencies. Fig. 7. Input PDFs or the Monte Carlo evaluation o the uncertainty o the sensitivity. The PDFs were created with.0 0 6 events. The histograms were accumulated with a binsize o 00. According Fig. 7 the symmetric distributions o the accelerations are dierent. In the case o the PDF obtained or the acceleration o the adapter the inluence o the rectangular distribution can be seen. The reason is that the standard deviation derived rom this measured points are less than in the case o the acceleration measured on the top mass. The physical origin o this behaviour is the dierent tendency to undergo rocking motion. The adapter is compact concentrated in the middle o the excitation axis, whereby the top mass can be more easily excited by transverse motions. The distribution o the orce signal is already converging a Gaussian distribution due to the highest number o measurements (37). The MC simulation was perormed in such a way, that irst all input PDFs or a certain requency were calculated with.0 0 6 events. Ater there, the output PDF was calculated according eq.. The calibration actors or the ampliier and the accelerations were taken with unity and the uncertainties connected with the calibration (bridge ampliier and vibrometer) were taken into account by adding a rectangular distribution to the input t-distributions o the voltage and the accelerations. From each output PDF the mean value and standard deviation were calculated (see Table ). In addition a discrete representation G o the output distribution unction G Y (η) (whereby η is the variable describing the possible values o the output quantity) or the output quantity Y was constructed according GUM upplement. This distribution can be used to determine a coverage interval in such a way, that with a certain probability p, all output values can be ound in this interval. By using this interval also the coverage actor can be obtained; i the coverage interval is divided through the standard uncertainty obtained rom the output PDF. In Figure 8 examples o output PDFs or dierent requencies are plotted. According this plot, one can see a shit o the distributions as a unction o the requency which is due to the loss o sensitivity with increasing requencies. In addition an increasing broadening o the distributions can be seen at higher requencies. The main reasons are mechanical inluences like rocking modes, which lead to a higher standard deviations o the acceleration input PDFs. In Table the uncertainty analysis according the standard GUM procedure is compared with the Monte Carlo method o GUM upplement. Thereby in the columns -3 the measured sensitivities with their dedicated relative uncertainties determined according the standard GUM are given. The partial dierentiation o the model (eq.) was done with the ree available sotware Maxima [4] to get the sensitivity coeicients. The sotware provides a Computer Algebra ystem which can be lexiblee linked to any programming language. The results o the sotware were validated punctual with the commercial sotware GUM-Workbench [5]. In column 4 and 5 the mean value and the relative standard uncertainty based on a coverage interval o 95,45% are given. In addition the coverage interval was used to calculate the k-actor, which is given in the last column. In column 6 the relative deviation o the measurand (col. ) and the mean value based on the MC simulation (col. 4) are given. This dierence is very small which is a proo that the number o trials (one million) used in the MC procedure is quite suicient. In column 7 the relative deviation o the relative uncertainties is shown; also based on the standard GUM value o column 3.
3 4 5 6 7 8 Frequency -Measure d/ -MC d/ (d/) k k (Meas.-MC) (Meas.-MC) MC Hz mv/v/n % mv/v/n % % % 8,75 4,60E-04 0,43 4,600E-04 0,5 0,04-9,04,80 5 4,639E-04 0,46 4,640E-04 0,55-0,0-8,70,83 3,5 4,645E-04 0,59 4,650E-04 0,68-0, -6,33,93 37,5 4,649E-04 0,44 4,650E-04 0,5-0,0-9,03,80 56,5 4,638E-04 0,44 4,640E-04 0,5-0,04-9,05,80 6,5 4,637E-04 0,44 4,640E-04 0,5-0,06-9,03,80 87,5 4,635E-04 0,50 4,630E-04 0,59 0,0-8,0,87 00 4,635E-04 0,67 4,640E-04 0,77-0,0-4,40,96 5 4,63E-04 0,50 4,630E-04 0,59 0,03-8,7,87 50 4,636E-04 0,46 4,640E-04 0,55-0,08-8,66,83 00 4,64E-04 0,44 4,640E-04 0,5 0,03-8,87,80 50 4,66E-04 0,44 4,630E-04 0,5-0,09-9,00,80 300 4,64E-04 0,44 4,640E-04 0,53 0,04-8,8,8 400 4,69E-04 0,45 4,630E-04 0,54-0,03-8,67,83 500 4,607E-04 0,46 4,60E-04 0,55-0,06-8,45,83 650 4,59E-04 0,73 4,590E-04 0,8 0,04-3,0,97 800 4,597E-04 0,49 4,600E-04 0,57-0,07-7,65,86 900 4,60E-04 0,46 4,60E-04 0,54 0,00-8,00,83 50 4,50E-04 0,55 4,50E-04 0,63-0,0-5,7,9 500 4,449E-04 0,7 4,450E-04 0,80-0,0 -,0,97 600 4,489E-04 0,77 4,490E-04 0,85-0,0 -,4,98 700 4,448E-04,03 4,450E-04, -0,04-8,60,00 800 4,354E-04,03 4,350E-04, 0,09-8,9,00 900 4,337E-04,33 4,340E-04,43-0,06-7,7,0 Table Comparison o the sensitivity and their uncertainty obtained with the standard GUM procedure and the MC-method.
Here one can see that, the uncertainty according the MC method is roughly 0-0 % higher than the values obtained with the conventional standard GUM method. The reason is that in the standard GUM approach the standard uncertainty o the mean value is calculated by the division o the standard deviation with the square root o the number o measurements. I only a limited number o measurements can be perormed, which is normally the case, rather the scaled and shited t-distribution must be applied, where the standard uncertainty u o the distribution is calculated rom the standard deviation s by: u n n 3 s n (8) Only in the case o ininite number o measurements this uncertainty converges to the Gaussian case. 5. CONCLUION The requency dependent sensitivity is the main output o a periodical orce calibration. Mostly adapters have to be used or coupling test masses to the transducer. The proper sensitivity can only obtained i the behaviour o such adapters is taken into account during the calibration. The main uncertainty contribution comes rom the acceleration o the test mass. Comparisons between CEM and PTB show a good agreement or the sensitivity or requencies up to khz. The results o the uncertainty evaluation with the standard GUM approach and the MC method deviate rom each other by 0 %- 0 % which is due to the assumption o dierent input distributions. All together one can reach uncertainties o % or requencies below.5 khz. ACKNOWLEDGMENT The EMRP is jointly unded by the EMRP participating countries within EURAMET and the European Union. REFERENCE [] C. chlegel, G. Kiekenap, B. B. Glöckner, A. Buß, R. Kumme Traceable periodic orce calibration, Metrologia 0, 49, n. 3, 4-35. [] European Metrology Research Programme (EMRP), http://www.emrponline.eu. [3] C. chlegel, G. Kieckenap, H. Kahmann, R. Kumme Mechanical Inluences in inusoidal Force Measurement, ACTA IMEKO, January 04, Volume 3, Number. [4] http://maxima.sourceorge.net/ [5] http://www.metrodata.de/