ERRORS AND UNCERTAINTIES IN EXPERINMENTAL MEASUREMENTS ELANKUMARAN NAGARAJAN 4TH DECEMBER 2011 SUMMARY Experimentation is an important process that will be carried out on new theories, ideas, products and devices to validate them. Despite the quality of the experiment there will always be some uncertainties associated with the experimental data. These uncertainties arise due to a random or systematic error. The uncertainties associated with a simple cantilever beam experiment, in which the deflection of the beam due to the applied load is recorded with both a dial gauge and a potentiometer.
With the most suitable measurements of beam deflection the average value of young’s modulus and the uncertainties associated with that is calculated. The results were in accordance with the theoretical predictions. The experiment revealed that by statistical analysis of the dataset and by repeating the experiments the random errors could be minimized. Index Page number •Introduction 3 •Theory 3 Experimental method 4 •Results 5 •Discussion 6 •Conclusion 7 •Reference 7 Introduction Experimentation is the most important process in engineering and science. Any ideas, theories, products and devices are validated only by experimentation.
Despite the quality of the equipments and the efforts of the experimentalists there will always be some uncertainties (variability) associated with the collected data. It is impossible to make exact measurements. The uncertainties arise due to two errors – random errors and systematic errors. Random errors are always present and they are random in nature. Systematic errors arise due to the technique adopted to perform the measurements. Random errors can be revealed by repeating the measurements and by statistical analysis of the dataset.
Systematic errors always push the results in the same direction and are often more difficult to identify. The aim of this experiment was to identify the uncertainties associated with a simple cantilever beam bending experiment. This was achieved by analyzing the geometry of the experiment, identifying the critical parameters affecting the results based on the simple beam bending equation, by repeating the measurements and by comparing the experimental data with the theoretical predictions. Theory A cantilever beam (figure 1); under the action of a concentrated load applied at its free end the beam will deflect.
Provided the applied load acts in the XY plane and deflection occurs in this same plane (figure 2), the magnitude of the deflection is given by the equation where: figure 1. a cantilever beam figure 2. the deflection of the cantilever beam figure 3. critical dimensions for the calculation of the second moment of area of a rectangular cross-section. In case of a beam of rectangular cross-section the second moment of area with respect to the axis m (figure 3) is given by the expression where: b is the width of the beam is the height of the beam Experimental method The end of a steel beam was clamped to an aluminium extrusion frame. The free end of the beam had been drilled to allow a set of nine weights to be hanged from that. The critical dimensions of the beam (L, X, b and h) were measured and the uncertainty associated with those estimates arising from the instrument used for measurement was noted. The dimension of L was 0. 30 m, X was 0. 27 m, b was 0. 0204 m and h was 0. 0029 m. Nine weights (99. 9 g, 101. 1 g, 98. 8 g, 99 g, 101 g, 100. 4 g, 99. 4 g, 103. 7 g and 99 g) were used.
The deflection of the beam under these weights was measured using two sensors – a dial gauge and a potentiometer wired to a digital display. The dial gauge and the potentiometer were zeroed while the beam was unloaded. The weights were added one by one to the free end of the beam and at each stage the mass added to the free end of the beam, the deflection readings obtained from both the potentiometer and the dial gauge were recorded. After adding all the masses to the free end of the beam, slowly the masses were removed one by one until the beam was subjected to no external loads.
The experiment was repeated five more times and the data were recorded at each stage in the same manner as the previous step. Results The displacement of the beam recorded by the potentiometer due to the applied force in six experiments is shown in the graph form in figure 4. figure 4. the deflection of the beam recorded by the potentiometer versus the applied force The displacement of the beam recorded by the dial gauge due to the applied force in six experiments is shown in the form of graph in figure 5. figure 5. he deflection of the beam recorded by the dial gauge versus the applied force From the set of data recorded from the dial gauge (which represent the beam deflection better) the young’s modulus (E) of the beam and the standard error of young’s modulus were calculated for each experiment and are presented in table 1. ExperimentsExp 1Exp 2Exp 3Exp 4 Exp 5Exp 6 E (GPa)205. 2206. 3207. 8207. 6206207. 5 Std error of E (GPa)2. 31. 81. 71. 71. 61. 4 From the recorded data and the calculated values the true value of young’s modulus is 206. GPa and the standard error associated with this is 1. 7 GPa. Discussion The results obtained were as expected. They were in accordance with the theoretical predictions. The difference in data in different experiments can be explained. The surroundings, technique adopted and the physical set up of the experiment were the reasons for the inconsistent data. The physical set up of the experiment was very much exposed to the surroundings, so that even a vey minute disturbance to the physical set up had its impact in the reading.
The inconsistency in the technique adopted through out the experiment was another main reason for the different readings. Figure 5 showed that the readings of the potentiometer were slightly differing from the theoretical prediction. That was due to certain systematic problem in the potentiometer itself. The results could have been improved with ideal experimental conditions and a more effort in maintaining the consistency of the technique adopted. Conclusion The true value of young’s modulus was calculated as 206. 7 GPa and the uncertainty associated with was calculated to be 1. GPa. The experiment revealed that uncertainties associated with experimental data are always present and it should be taken into account, as they can affect the results. The experiment should carried out with quality equipments, ideal lab surroundings and best effort from the experimentalists. Reference •Gere, J. M. and Timoshenko, S. P. mechanics of the materials third edition, Chapman & Hall (1991) •Kirkup, L. Experimental methods: an introduction to the analysis and presentation of data, Wiley (1994) •Taylor, J. R. An introduction to error analysis, university science books (1997)