ERRORS AND UNCERTAINTIES IN EXPERINMENTAL MEASUREMENTS
ELANKUMARAN NAGARAJAN
4TH DECEMBER 2011
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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
2
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:
3
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
h is the height of the beam
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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.
0.009
0.008
0.007
pot 1 (m)
0.006
pot 2 (m)
0.005
pot 3 (m)
0.004
pot 4 (m)
pot 5 (m)
0.003
pot 6 (m)
0.002
theory (m)
0.001
0
0
2
4
6
8
10
5
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.
0.009
0.008
0.007
dial 1 (m)
0.006
dial 2 (m)
0.005
dial 3 (m)
0.004
dial 4 (m)
dial 5 (m)
0.003
dial 6 (m)
0.002
theory (m)
0.001
0
0
2
4
6
8
10
figure 5. the 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.
Experiments Exp 1
Exp 2
Exp 3
Exp 4
Exp 5
Exp 6
E (GPa)
205.2
206.3
207.8
207.6
206
207.5
Std error of
E (GPa)
2.3
1.8
1.7
1.7
1.6
1.4
From the recorded data and the calculated values the true value of young’s
modulus is 206.7
GPa and the standard error associated with this is 1.7 GPa.
Discussion
6
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 1.7 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)
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