EXPRESSION OF MEASUREMENT UNCERTAINTY IN TEACHING OF MECHANICAL ENGINEERING

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1 15 th International Conference on Experimental Mechanics PAPER REF: 3095 EXPRESSION OF MEASUREMENT UNCERTAINTY IN TEACHING OF MECHANICAL ENGINEERING Edivaldo A. Bulba 1(*), Helena V.G. Navas 2 1 Fundação Educacional Inaciana (FEI), São Bernardo do Campo, São Paulo, Brazil 2 UNIDEMI, Departamento de Engenharia Mecânica e Industrial, Faculdade de Ciências e Tecnologia, FCT, Universidade Nova de Lisboa, Campus de Caparica, Portugal (*) ebulba@gmail.com ABSTRACT The teaching and learning processes of the expression of measurement uncertainty is an arduous task. It covers several concepts of calculus, statistical inference, as well as concepts inherent to the own experience in metrology. Thus, this set of information makes difficult its assimilation at the beginning of metrology learning. This work highlights how it is possible to clarify several of these concepts through simulation and software application in parallel with the formal education of measurement uncertainty expression, improving the student's learning in the field of metrology. Keywords: uncertainty, measurement uncertainty, teaching, mechanical engineering INTRODUCTION The execution of a measurement provides the quantification of a given magnitude based on a reference scale. The parameters of this quantification and the comparison between measured and nominal values are metrology's issues. Metrology is essential in scientific research. The measurement science is of growing importance (Howarth, 2004). All fields of human activities need to carry out measurements of specific parameters inherent to each activity. The metrology is a basic tool for industry and industrial applied scientific research. The metrology education is getting importance in the curricula of engineering and technological university courses. The basic problem of any quantitative experience is the identification of the true value of measured magnitude. The improvement of a measurement system means the convergence of the measured value to the true value. The true value is never achieved, as the Zeno s paradoxes. All means of measurement follow a statistical distribution where the true value lies within a certain degree of confidence (Heisenberg Uncertainty Principle). The error is the difference between the measured value of a quantity and a reference value (IPQ, 2008). In the past, the error was considered the most important aspect of measurement (IPQ, 1998). Currently, the quantification and definition of uncertainty and its influence of the measurement have been the fundamental targets of metrology. The uncertainty is a parameter associated with the result of a measurement; it characterizes the dispersion of values that can be attributed to a measurand (IPQ, 2008). The scientific basis of metrology is a measurement uncertainty. Uncertainty of measurement comprises different components. ICEM15 1

2 Porto/Portugal, July 2012 The Guide to the expression of uncertainty in measurement, commonly known as ISO-GUM (JCGM100, 2008) is the more important document in this field. The discipline of metrology, present in many Mechanical Engineering courses curricula, becomes incomplete without the method described by the ISO-GUM. In accordance with the classical ISO-GUM method, there are following steps to obtain uncertainty in measurement: 1. Definition of the mathematical model of measuring; 2. Definition of uncertainty components; 3. Estimation of standard uncertainties; 4. Calculation of sensitivity coefficients; 5. Evaluation of possible correlations between the components of uncertainty; 6. Definition of combined standard uncertainty; 7. Calculation of effective degrees of freedom effective; 8. Obtaining of expanded uncertainty. Uncertainty of measurement according to ISO-GUM is a very robust tool and of great practical importance, has been applied by a growing number of companies and institutions in the customer-supplier relationship, thus justifying its teaching in engineering courses. However teaching the ISO-GUM together with the discipline of metrology in undergraduate courses becomes an arduous task, because in addition to requiring from the student a good basis of calculation and statistical inference, there is the problem of the scant time to address several fundamental concepts inherent to metrological terminology contained in two related documents: International Vocabulary of Metrology (VIM) and ISO-GUM. Nevertheless, there are software applications to facilitate this teaching experience. TEACHING OF ISO-GUM WITH THE AID OF SOFTWARE There are basically two methods for obtaining the measurement uncertainty: the classical method (CM), whose steps are listed above and involve a considered number of calculations and the method of Monte Carlo simulation (MMC). The computer programs that employ the CM tend to dramatically reduce the time devoted to calculations concerning several steps until you reach the value of expanded uncertainty, such as calculations of partial derivatives as sensitivity coefficients, effective degrees of freedom, among others. This advantage is evident when we put a relatively small amount of input data and obtain easily the expression of measurement uncertainty. The initial focus of the learning is on the correct interpretation of data input and output. This procedure just described for using the CM, can be used with MMC too. Usually the CM is used for conventional problems of measurement uncertainty, where the mathematical model is linearizable and the probability density function of the output data describes a distribution close to normal distribution. In the absence of these conditions, the MMC can be used. However, nothing prevents us to use the MMC where the classical or traditional method also works, moreover the supplement that describes this method (JCGM 101:2008), encourages 2

3 15 th International Conference on Experimental Mechanics such can be used in conjunction with the traditional method in all circumstances because it is an experimental method. The main advantages of the MMC in terms of didactic point of view: 1. It is a practical method, it is easy to understand. The result of the simulation by Monte Carlo model is a propagation of distributions. This method is more intuitive than the traditional method of uncertainty propagation. Until a few years ago, the speed of computers was a restriction on the MMC application because it was too slow to simulate a sufficient number of iterations in order to trust in the result and to be compatible with the CM. However, currently this problem is overcome. The possibility to observe the distribution of the response variable facilitates the interpretation of concepts related to the measurand. " illustrating the distribution of the resulting uncertainty for each measuring, obtained from a Monte Carlo simulation, allow expressing the measurement uncertainty of richer way, which can more easily understand the meaning of controversial terms: output estimate, uncertainty, spanning range and level of confidence (probability of comprehensiveness). Graphics illustrating the distribution of the resulting uncertainty for each measurand, obtained from an appropriate Monte Carlo Simulation, allow expressing the measurement uncertainty in a richer way. It can help to more easily understanding of the meaning of some controversial terms: output estimate, expanded uncertainty, and range of coverage and confidence level (coverage probability). The use of specialized software allows displaying such information without any additional work for the Metrologist (Aragón, Sanches, 2008). 2. One of the steps of defining the measurement uncertainty is getting the sensitivity coefficients estimated by calculating the partial derivatives. Particularly in complex mathematical models that can be very labor intensive. In this case the MMC greatly simplifies because with it comes to measurement uncertainty without the calculation of derived. Although the procedure by MMC does not include the sensitivity coefficients, because it fixes all input variables but one, that is, by varying only one at a time, we can determinate the relation between the variation of input with variation of output which result can be taken as a coefficient of sensitivity more accurate than that obtained by partial derivatives by traditional method because such coefficient includes not only the first order Taylor series, but all higher-order terms (JCGM 101: 2008). Thus we can evaluate the influence of each input uncertainty on the combined standard uncertainty and this is a practical way for the student to understand the meaning of these coefficients. 3. The same applies to the sensitivity coefficients, the effective degrees of freedom by Welch-Satterthwaite are not calculated when MMC is used, which also contributes to the simplification of obtaining the expanded measurement uncertainty. In parallel with MMC we also use the CM that performs automatically the procedures listed above. For conventional problems of uncertainty, both methods produce very similar results. All advantages provided by both methods, with emphasis on employment of MMC, is motivating because the student begins to assimilate the concepts behind the expression of measurement uncertainty. When this step is consolidated, we can follow with the intermediate steps to obtain the uncertainty. ICEM15 3

4 Porto/Portugal, July 2012 SOFTWARE APPLICATION EXAMPLE The example below (Oliveira, 2008) refers to the expression of uncertainty of measurement of Brinell hardness. F = 3000 ±45kgf (force applied by penetrator) The expanded uncertainty of ± 45kgf was given by the hardness meter, hence getting the standard uncertainty D = 10mm ±0,010mm (diameter of penetrator ball) The expanded uncertainty of ± 0,010mm is given by the supplier in your calibration certificate, hence getting the standard uncertainty d = 4,15mm and (diameter of the spherical cap), these values of mean and standard deviation obtained from six trials with measurements 4,10; 4,15; 4,20; 4,15; 4,10 e 4,20mm, hence the standard uncertainty Applying a software that performs both the MMC and CM (Gum Workbench, 2010), we enter the expanded uncertainty values of F and D, the sample with the 6 d measurements and the mathematical model of hardness HB. With only this information the program issues a comprehensive report about the measurement uncertainty, with many important information: The nominal value of The partial derivatives: 4

5 15 th International Conference on Experimental Mechanics \derive\hb\sl\\derive\f = 2.0/( D (D - sqrt(d^2.0 - d^2.0))); \derive\hb\sl\\derive\d = (-2.0 F ( D ( D/sqrt(D^2.0 - d^2.0)) + (D - sqrt(d^2.0 - d^2.0)) ))/sqr( D (D - sqrt(d^2.0 - d^2.0))); \derive\hb\sl\\derive\d = (-2.0 F D (-0.5 (-2.0 d)/sqrt(d^2.0 - d^2.0)))/sqr( D (D - sqrt(d^2.0 - d^2.0))); Their respective sensitivity coefficients 0,071; 2,1e The standard uncertainties of 22,5kgf; 0,005mm and 0,0183mm. The contributions of input uncertainties on absolute values and percentages. The effective degrees of freedom by Welch-Satterthwaite are equal to 13. The expanded uncertainty of,6kgf/mm ², at a level of confidence of 95.45%. Then with the same software we use the MMC, to solving the same problem, with two million of iterations. The results are equivalent to the CM results: Nominal value of Expanded uncertainty of 5, 9kgf/mm² at 95.45% level of confidence. It also offers a histogram by comparing the values of average and expanded uncertainty obtained by classical methods (dashed red lines) and MMC (dashed blue lines): Fig. 1 Histogram obtained by MMC ICEM15 5

6 Porto/Portugal, July 2012 RESULTS AND CONCLUSIONS The teaching of uncertainty in measurement is permeated by several metrological, statistical and terminological concepts. The calculations are very hard (for example, a partial derivatives calculation or calculation of effective degrees of freedom). There are software applications available in educational versions that could make the calculations far easier. The usage of simple computer simulations based on probability distributions of measuring also can contribute to a more effective assimilation of metrology concepts. ACKNOWLEDGMENTS One of the authors (HN) would like to thank the Faculdade da Ciência e Tecnologia da Universidade Nova de Lisboa (FCT-UNL) and the Fundação para a Ciência e Tecnologia (FCT) for their support. REFERENCES Aragão e Silva BJG, Sanches E. Cálculo de Incerteza em Ensaio de Tração com os Métodos de GUM Clássico e de Monte Carlo. ENQUALAB Congresso da Qualidade em Metrologia. REMESP. 09 a 12 de Junho, São Paulo, Brasil, 2008 GumWorkbench, Howarth P., Redgrave F. Metrology in short. MKom Aps: Denmark. 2nd ed. May, 2004 IPQ, GUM - Guia para a expressão da incerteza de medição nos Laboratórios de Calibração, IPQ, 1998 IPQ, VIM Vocabulário Internacional de Metrologia, Guia ISO/IEC 99, Versão Portuguesa, 3ª Ed., IPQ, 2008 JCGM 100:2008 Evaluation of measurement data - Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections) JCGM 101:2008, Evaluation of measurement data -Supplement 1 to the Guide to expression of uncertainty in measurement -Propagation of distributions using a Monte Carlo method, 2008 Oliveira JEF. A Metrologia Aplicada aos Setores Industrial e de Serviços, SEBRAE,