MAE 123 : Mechanical Engineering Laboratory II …jmmeyers/ME123/Lectures/ME123 Lecture 1...MAE 123...

MAE 123 : Mechanical Engineering Laboratory II

Introduction to Experimental Practice


Introduction to Experimental PracticeDr. J. M. Meyers | Dr. D. G. Fletcher | Dr. Y. Dubief

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I. Error

Illegitimate error

Random error

Systematic error

II. Accuracy and Precision

Accuracy

Precision (Relative and Absolute)

Significant Digits

III. Propagation of Error: Uncertainty

IV. Parameter Sensitivity

by differentiation

by logarithmic differentiation

V. Presentation of Scientific Data

OUTLINE

2



Error: The difference between the true value and a

calculated or observed (measured) value.

There are three types of error we will consider

a.Illegitimate error: a mistake is made in procedure,

leading to a bad value

b.Systematic error: an often unintentional bias that

enters into observations owing to how instruments are

read, faulty calibration of instrument, etc.

c.Random error: fluctuations in observations from

experiment to experiment

I. ERROR

3



ILLIGITIMATE ERROR

Error resulting from a careless mistake made in procedure,

leading to a bad value.

I. ERROR

Illegitimate errors can be corrected by recognizing the

mistake and redoing the measurement.

4

Illegitimate Error Examples:

• misreading a ruler

• copying down a wrong number

How to Minimize:

Just be more careful!



SYSTEMATIC ERROR

I. ERROR

Systematic errors can be eliminated only by critical examination

of the experimental method, and often this is accomplished by

performing an experiment with the apparatus to measure a

known or analytically determined value. Another approach is to

have a different person perform the same experiment!

5

Systematic Error Examples

• The cloth tape measure that you use to measure the length of an object had been stretched out

from years of use. (As a result, all of your length measurements were too small.)

• The electronic scale you use reads 0.05 g too high for all your mass measurements (because it is

improperly tared throughout your experiment).

• Using a timer that is slow, so that all the times measured are slow by the same factor.

How to minimize

Systematic errors are difficult to detect and cannot be analyzed statistically, because all of the data is

off in the same direction (either to high or too low). Spotting and correcting for systematic error takes

a lot of care.

• How would you compensate for the incorrect results of using the stretched out tape measure?

• How would you correct the measurements from improperly tared scale?

Owes to unintentional bias that enters into observations due to

how instruments are read, faulty calibration of instrument, etc.

A reproducible error that biases the data in a given direction.



Fluctuations in observations from experiment to experiment

that follow statistical distributions.

I. ERROR

Random errors are not something that can be controlled and

are typically present owing to limitations on instrumentation

or to an inability to completely control the test conditions.

6

Random Error Examples:

• You measure the mass of a ring three times using the same balance and get slightly different values:

17.46 g, 17.42 g, 17.44 g

• Noise in an oscilloscope reading

How to minimize or eliminate random errors:

• Take more data. Random errors follow statistical distributions thus can be evaluated through

statistical analysis and can be reduced by averaging over a large number of observations.

RANDOM ERROR

Random error is not the fault of you or the equipment



ACCURACY: a measure of how close the experimental (or

calculated) value is to the true value

PRECISION: a measure of how exactly the experimental

result is determined without reference to what the result

means. Precision is also a measure of how reproducible the

result is.

Absolute precision: indicates magnitude of uncertainty in

the same units as the result

Relative precision: indicates the uncertainty as a fraction

of the value of the result.

From the definitions it is clear that accuracy ≠ precision

The precision of an experimental observation determines the

number of significant digits used to report the value -- points

will be deducted for using too many digits to report an

answer.

II. ACCURACY AND PRECISIONII. ACCURACY AND PRECISION

7



SIGNIFICANT DIGITS:

a. Left-most non-zero digit is most significant digit

b. If no decimal point, right-most non-zero digit is least

significant digit

c. If decimal point right-most, digit is least significant even if 0

d. All digits between least and most significant count as

significant digits

8

•Reporting in scientific notation makes it easier to keep track of significant digits in calculations

•Quoting result of experiment should use one figure more than determined by experimental

precision

•In calculations keep this extra digit for each variable, so that the final result retains equivalent

precision from all input

II. ACCURACY AND PRECISION

1,234

123,400

123.4

1,001

1,000.

10.10

0.0001010

1.234(10)3

1.234(10)6

1.234(10)2

1.001(10)3

1.000(10)3

1.010(10)1

1.010(10)-4

All numbers below have 4 significant

digits by the above rules.

In many cases it is beneficial

to use scientific notation

How many significant

digits does this readout

have?

What would be its

representation in

scientific notation?



A student measures table top with steel meter rule and finds an average value of 1.982 m for the

length.

This student finds after that the rule was calibrated at 25 C, and has an expansion coefficient of

0.0005/C.

The experiment was done at 20 C, so the student multiplies result by 1-5(0.0005) = 0.9975, so his

result is now 1.977 m.

This illustrates a systematic error due to incorrect calibration and how this particular systematic

error can be corrected.

The table manufacturer lists the table at a length of L = 2.000 m. This means the result is

inaccurate by 23 mm.


9

Example 1: Tabletop Measurement Error, Precision, and Accuracy

Precision: reported length is precise 1.977 m.

Represented significant digits Imply measurement has an absolute precision of 1 mm. (relative

precision on the order of 1/1977)



Additional Notes:

• Round off the calculated result from all input to the reflect

the precision, and use the normal rounding rules (if greater

than 1/2 round up, less round down, equal to 1/2 round to

even)

• Accuracy depends on how well we control systematic error

(reproducible differences)

• Precision depends on how well we handle random errors

How can these be managed?

• Improvements to instrumentation (if you can afford it) if it is

due to imprecise instrumentation

• If the random errors come from imprecise control of

conditions for the experiment, then we can reduce the errors

by taking more measurements over a longer period, and the

effect of random errors will be reduced on the average

result.

10




•Generally, the result of our experiment is an estimate of the true value.

•We can only estimate the errors since we often do not know the true value in advance

•Repeating experiment often gives a different result, so we look to see if the results agree within

estimated uncertainty.

•Repeating the experiment to reduce influence of random errors means we will usually estimate a

mean and standard deviation for the data -- statistical analysis

Usually, the result we are looking for in an experiment is determined from a combination of

measurements. Thus, the uncertainty in the result is determined by the uncertainty in the separate

measurements. What is this dependence?

11

III. PROPAGATION OF ERROR: UNCERTAINTY



� � = 12�� exp − � − �

2��

Measured data with only random error contributions follow a normal (aka Gaussian) distribution with

parameters and �. This probability density function is given by:

12


�� ≡ variance� ≡ standarddeviation ≡ meanvalue

FWHM = 2σ −2ln 1/2-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

1

2

3

4

FWHM

FWHMf(x)

x

µ = 0, σ = 0.1 µ = -1, σ = 0.1 µ = 0, σ = 0.5

FWHM



• Normally, measurements of more than one parameter are made to determine or reduce a

property of interest.

• Each of these individual parameter measurements will have their own respective random error.

• In many experiments it is important to understand how these measurement errors affect the

overall uncertainty of these other properties.

• Consider the problem of computing the quantity # which is a function of $ independent variables

to be measured: �%, ��, �', … �)* = +(-., -/, -0, … -1)

13

�̅% =145 �% 6

6

�̅� =145 �� 6

6

�̅' =145 �' 6

6

�̅) =145 �) 6

6

�78� =1

4 − 15 �% 6 − �̅% �9

6:%

�7;� =1

4 − 15 �� 6 − �̅� �9

6:%

�7<� =1

4 − 15 �' 6 − �̅' �9

6:%

�7=� =1

4 − 15 �) 6 − �̅) �9

6:%


The average or mean values of the

measured quantities are:

The corresponding variances:

(recall variance = (std dev)2)


Introduction to Experimental Practice14

#6 − #̅ ≈ �% 6 − �̅%?#?�% 7;,7<,…7=

+ �� 6 − �̅�?#?�� 78,7<,…7=

+ �' 6 − �̅'?#?�' 78,7;,…7=

+⋯+ �) 6 − �̅)?#?�) 78,7;,7<,…7=B8

Each combination of measurements can be used to estimate the deviation from the average value

using a Taylor Series expansion:

For illustration simplicity, let’s assume our desired parameter # is a function of just two measured

quantities, �% and ��:


#6 − #̅ ≈ �% 6 − �̅%?#?�% 7;

+ �� 6 − �̅�?#?�� 78



If we square both sides and sum over all measurements, then we can replace the difference terms by

variances (��). How many terms are there on RHS?


�D� = �78�?#?�% 7;

�+ 24 − 15 �% 6 − �̅%

?#?�% 7;

�� 6 − �̅�?#?�� 78

9

6:%+ �7;�

?#?�� 78

�

Here we usually assume that the two variable “errors” are not related -- that they are statistically

independent. This is the most common situation. If a large enough sample set is then taken, any

covariance term will be zero:

�D� = �78�?#?�%

�+ �7;�

?#?��

��D = �78�

?#?�%

�+ �7;�

?#?��

�

=0

But what about calculated parameters that are a function of more than two measurements?

Thus only two RHS terms will remain. With the assumption of independence our estimate of the

variance of the derived result is then:

5�7=�7E9

%= 0



In fact, it can be shown that for a desired parameter that is a function of $ number of measured

quantities:


# = �(�%, ��, �', … �))

�D = �78�?#?�%

�+ �7;�

?#?��

�+ �7<�

?#?�'

�+⋯+ �7=�

?#?�)

�

that all the cross terms will cancel and the total uncertainty of parameter # can then be derived from

the following relation:

Let’s apply this to a couple of examples...

Error in

measurement

of �%

Error in

measurement

of ��

Error in

measurement

of �'

Error in

measurement

of �)

Partial derivatives of calculated parameter # with respect to individual measured parameter while

holding other measured parameters constant under assumption of statistical independence

General Form for

the Expression of

Uncertainty




We now want to estimate how far we probably are from this value owing to the imprecision of our

measurements (we hope we have eliminated systematic errors and that we are only dealing with

random errors -- we checked the thermometer and pressure transducer for proper calibration.)

G = HIJWe want to determine the density of room air, which is very difficult to measure directly. We recall

the ideal gas law:

H = GIJ

As we are working with air, we know that:

I = IK)6L/(MW)= 8.3145 N-m/(mol K)/0.029 kg/mol = 287.0 (N-m)/(kg-K).

We take turns reading the thermometer and find an average value for room J of 298 K, with a

standard deviation of �M = 2 K.

We also use a pressure transducer to measure the ambient condition, and after repeating this many

times find an average value of G = 101000 N/m2 with a standard deviation of �N= 100 N/m2.

H = GIJ =

101000(287.0)(298) = 1.18kg/m

3

Example 2: Measurement of Density (1/2)




We have only two measurements in calculating H (pressure, G, and temperature, J). Using the

derived general form for the expression of uncertainty (Slide 15) we obtain:

�V = �M�?H?J

�+ �N�

?H?G

�

?H?J = −

12GIJ�

?H?G =

1IJ

�V = �M� −12GIJ�

�+ �N�

1IJ

�

�V = 2� −12101000

287.0(298)��+ (100)� 1

287.0(298)�

�V =4.13(10)-3 kg/m3

H = 1.18 ± 4.13 10 −3 kg/m3

Example 2: Measurement of Density (2/2)

The partial differentials of each measured quantity while holding the other measure quantity as

constant are:

Substituting these relations into our relation for uncertainty:

Adding our measured values and working through we obtain:

or




G + 12HY� + HZ[ = constant

The Bernoulli principle is a very useful concept in fluid dynamics because it is easy to use and can give

great insight into the balance between pressure, velocity and elevation. Simply stated, at any point

along a streamline of fluid flow, the summation of pressure, dynamic pressure, and head pressure will

be constant.

Assuming the two measurement points (1 and 2) are at the same elevation ([% ≈ [�):

G% +12H%Y%

� = G� +12H�Y�

�

Example 3: Pitot Probe Measurement (1/3)

This is the key relation in extracting fluid velocity from an instrument called a Pitot probe.





G% +12H%Y%

� = G� +12H�Y�

�

At the static pressure port (location 1), the velocity is normally taken to be 0 as this measurement is

perpendicular to the flow direction (ergo no dynamic pressure):

0

Solving for velocity at measurement point 2:

Y� = 2 (G% − G�)H�

It is common practice to measure a pressure

difference rather than the two pressure values,

thus:

Y� =2∆GH�





We have only two measurements in calculating H. Using the derived general form for the expression

of uncertainty (Slide 15) for two variables:

?Y�?∆G =

12

2H�∆G

?Y�?H� = −

122∆GH�'

�D = �78�?#?�%

�+ �7;�

?#?��

� �K; = �∆N�

?Y�?∆G

�+ �V;�

?Y�?H�

�

Y� =2∆GH�

�K; = �∆N�12

2H�∆G

�

+ �V;�122∆GH�'

�

�K; = �∆N�1

2H�∆G + �V;� 2∆GH�'

and

We’ll leave the derivation here as is. But what

would happen if we were to insert the ideal

gas law for density as described in the

previous example? Would the we arrive at the

same uncertainty?



Parameter sensitivity by differentiation:

In our density calculation from Example 2 recall that:

These relations express the fractional change in density caused by a fractional change in the two

independent variables, G and J. These express the non-dimensional sensitivities of density to the

two variables.

To find the pressure sensitivity, we differentiate the expression w.r.t. G, keeping J constant, and

then w.r.t. to J keeping G constant.

Replacing the partial terms with the full differential, we can rearrange the two relations into more

convenient, intuitive forms:

22

IV. PARAMETER SENSITIVITY

H = GIJ

?H?G M

= 1IJ =

GIJ1G =

HG

?H?J N

= − GIJ� = −

GIJ1J = −

HJ

]HH =

]GG

]HH = −

]JJ



Now going back to Example 2 again our general relation for density depending on BOTH temperature

and pressure is:

Using the partial differentials from the previous slide and if we let:

Dividing both sides by density squared yields the following:

23



�V� = �M�?H?J

�+ �N�

?H?G

�

�V = ]H �N = ]G �M = ]J

]H� = ]J� −HJ�+ ]G� HG

�We then arrive at:

]HH

�= ]G

G�+ ]J

J�



Note:

1. The dependence on the temperature and pressure uncertainties is linear in each case

2. The contributions sum as squares and there is no cross term (we assume that pressure and

temperature are independent of each other)

3. Using the fractional representation of the variance keeps dimensions out of the relation, and

makes it a general result

24


]HH

�= ]G

G�+ ]J

J�

H = GIJ




We then differentiate term-by-term, and this produces the fractional variances directly, with the

proper sensitivity scaling.

Another approach provides a short cut for differentiation. We take the natural log of both sides of

the expression:

We now square both sides, and assume that the cross term is zero (recall the assumption of

independent measurements of the separate parameters, G and J). This yields the same result as

the formal approach:

25


Parameter sensitivity by logarithmic differentiation:

ln H = ln G − ln I − ln JH = GIJ

]HH =

]GG −

]JJ

]HH

�= ]G

G�− 2 ]G

G]JJ + ]J

J�

0

]HH

�= ]G

G�+ ]J

J�



V. PRESENTATION OF SCIENTIFIC DATA

Measurements are normally displayed and represented in Tables or Graphs.

Units of each listed value must be included.

0 50 100 150 200 250 300 350 400 450

20

40

60

80

100

120

140

160

180

Water Temperature Heater Temperature

Tem

pera

utre

[C

]

Time [s]

Time

Water

Temperature

Heater

Temperature

[s] [C] ±3 [C] ±10

30 17 120

60 18 123

90 20 125

120 22 128

150 24 132

180 25 120

210 24 142

240 22 135

270 18 141

300 16 145

330 20 147

360 22 154

390 24±4 152

420 21±1 157

Graphs:

• Both axes must include measurement label

and unit

• Error bars for each representative data point

must be graphically illustrated

• If more than one data record is being

represented, an appropriate legend must be

used.

• Scale axes accordingly to represent what you

are trying to illustrate with your graph

Tables:

• Each column must include a header

measurement label and unit

• Error can either be represented in the

column header (if universal)

• If error is not universal the value must be

represented for each data point



GLOSSARY OF TERMS

Accuracy (of measurement) Closeness of agreement between a measured value and a true value. The term "precision" should not be

used for "accuracy".

Decimal places -the number of digits to the right of the decimal point.

Discrepancy - A significant difference between two measured values of the same quantity.

Error (of measurement) The inevitable uncertainty inherent in measurements

Illegitimate error- mistake or blunder - a procedural error that should be avoided by careful

Law of propagation of uncertainty - shows how the uncertainties (not the errors) of the input quantities combine

Precision - The degree of refinement with which an operation is performed or a measurement stated [Webster].

Random error -Statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement

device.

Relative error- Error of measurement divided by a true value of the measurement. Relative error is often reported as a percentage.

Significant figures - all digits between and including the first non-zero digit from the left, through the last digit .

Systematic error -A reproducible inaccuracy introduced by faulty equipment, calibration, or technique. Unlike random errors,

systematic errors cannot be reduced by increasing the number of observations nor do they follow random error’s statistical trend.

True value (of a quantity) The value that is approached by averaging an increasing number of measurements with no systematic

errors.

Uncertainty (of measurement) Associated with the result of a measurement, that characterizes the dispersion of the values that could

reasonably be attributed to the measurand. The uncertainty generally includes many components which may be evaluated from

experimental standard deviations based on repeated observations.

Absolute uncertainty The total uncertainty of a value.

Relative (fractional) uncertainty - the absolute uncertainty divided by the measured value, often expressed as a

percentage.

MORE TO COME…



REFERENCES AND FURTHER READING

1) T. Arts, H. Boerrigter, J.-M. Buchlin, M. Carbonaro, G. Degrez, R. Denos, D. Fletcher, D. Olivari, M. L.

Riethmuller, and R. A. van den Braembussche, “Measurement Techniques in Fluid Dynamics,” von Karman

Institute for Fluid Dynamics, 2nd Revised Edition.

2) E. O. Doebelin, “Measurement Systems: Application and Design,” McGraw-Hill Book Company, 1983, New

York

3) H. H. Ku, “Notes on the use of propagation of error of formulas,” Journal of Research of the National Bureau

of Standards, 70C (4):262

4) P. R. Bevington, “Data Reduction and Error Analysis for the Physical Sciences,”

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MAE 123 : Mechanical Engineering Laboratory II …jmmeyers/ME123/Lectures/ME123 Lecture 1...MAE 123...

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