Ch4 Data Acquisition

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MEC 400 –

INTRODUCTION TO ENGINEERINGAND PROBLEM SOLVING

CHAPTER 4Data Acquisition anR!"#!s!ntation


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It is important for engineers to be able to collect, record, evaluate, and

interpret experimental data.

Engineers normally obtain data either directly from mathematical models or

indirectly by means of comparison involved in measurements, namely,

testing and experimentation.

Though, testing and experimentation may be used interchangeably, they

have some differences as will be explained in the next section. The data

obtained from these measurements are then used to quantify the physical

process by relating numerical values to physical parameters andvariables.


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Definitions for terms and terminologies related to data acquisition:

Measurement: a process of quantifying physical quantities using calibrated

instruments and established procedures to obtain numerical values of

varying degree called data.

Standard: a permanent or easily reproducible record of the quantity or

measurement

Calibration: the process of validating a measuring instrument into

agreement with the physical standards! upon which the instrument is

based. Parameter: any of a set of physical quantities whose value characteri"e

some aspect of the system or model and remains unchanged, at least for

the duration of the investigation

Variable: a quantity that may change during the course of investigation.

There are two types of variable: dependent and independent variables

Engineering testing: measurement conducted to validate the performance of

the system, process, or mathematical models which was done before.

Engineering experimentation: measurement to discover something new or

previously not #nown or to demonstrate something suspected.


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General Guidelines When Conducting Experiment

The following guidelines serve as a common rule when performing an

engineering experimentation or testing.

$ollow all laboratory procedures and precautions attentively and in some

cases, strictly. %ome of the laboratory equipment may be very delicate,

sensitive and expensive, so it must be handled with extreme care, or that

sometimes you may have to deal with ha"ardous materials. %o, do not play

around in the laboratory.

&se formal data sheets or noteboo#s to record all data and other

observations.


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General Guidelines When Conducting Experiment

'ecord all information about instruments and experimental apparatus used.

$or illustration purpose, s#etch the physical arrangement of the equipmentused.

(alibrate all your measuring instruments for its validity before you start

using them. 'ead or review equipment manuals for instrument accuracyand precision.

)a#e all measurements as accurately and reasonably as economics willallow. In certain cases only rough estimates are required since only aminimal accuracy in measurement is needed. *ote that, however, although

care has been exercised some degree of inconsistency will inevitablydevelop in all experimental wor# which necessitates the need of erroranalysis.


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+hen recording data, write down each entry clearly and eligibly. Do not

erase your data or observations, even if it is a mista#e. %imply cancels it by

drawing a line across it. This is ust in case when you want to refer to it later.

-ave your data verified before leaving the laboratory. ften your instructor

or lab assistant will be able to help you confirming your results. %ome data

sheet even requires an endorsement from the respective authority.


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$ormat of formal report

/. bective %ection0state aim and obectives clearly1. Equipment %ection0list equipment with s#etches and diagrams duly labeled

with figure number

2. Theory %ection0 brief mention of theory or related equations

3. 4rocedure or )ethodology %ection0write in steps how the experiment isconducted

5. Data %ection0properly rounded and use appropriate significant figures

6. (alculations %ection0sample calculation, error analysis in logical sequence

7. 'esults %ection0tabular, graphical or combination of both forms

8. (onclusion %ection0achievement related to obectives, practical application,validity, errors encountered, suggestions for improvement


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%uggestion format for this course

/. bectives of the Experiment

1. 'esults2. Discussions of 'esults

3. %ample (alculations

5. 9ppendices


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Empirical function are generally defined as those based on values $#o%!&"!#i%!nt an o's!#(ation #at)!# t)an t)!o#*.

)athematical expressions can be modeled to fit experimental functions

Types of empirical functions :

Lin!a# : y = mx + c +

;inear function will plot as a straight line.


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E&"on!ntia,: y = a e mx

-)!n ",ott! on s!%i,o. "a"!# /i,, '! ,in!a#

In ,o. $o#% ,o. ymx ,o. e 1 ,o. a

Po/!# : y = ax m

In ,o. $o#% ,o. ym ,o. x 1 ,o. a


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P!#ioic + y = sin Ө

repeats its values in regular intervals or periods. The most importantexamples are the trigonometric functions, which repeat over intervals of

length 1


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Different methods or techniques are available to arrive at the =best straight

line> fit.

Three methods commonly employed for finding the best fit are as below:

/. %elected points method1. 9verage method0line location is positioned to ma#e algebraic sum of

absolute values of the differences between observed and calculated

values of the ordinate equal to ?

2. ;east square method


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%elected points method

Determine equation that best fit data that exhibit linear relationship.

( y = mx + c )

%elect line that goes through as many data points as possible and hasapproximately the same no of data points on either side of line.

%elect two points on the line at a reasonable distance apart:

let@s say 9x/, y/! and Ax1, y1!, the slope m is determined as

/!

the y0intercept c may be either obtained directly from the graph or by

substituting point 9 or point A bac# into the equation. 'earranging the

equation y B mx C c in the form c B y mx and using point 9x/, y/! for

example, we have c B y/ mx/ 1!

12

12

x x

y y

m −

−

=


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%elected points method


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;east square method

nown as regression analysis.

'egression analysis is a study of the relationships among variables. It is

used to determine a curve that minimi"es the error and provides a goodapproximation to the physical problem

Linear regression simplest form! 0 using a straight line to fit a set of data

points x /,y /!, x 1, y 1!, ..., x n, y n!

where the dependent variable y and the independent variable x have linear

relationship. The equation of the straight line is

y = mx + c + e

c B y 0intercept

m B slope of the linear line

e B error between the measured y value and the approximate value given by equation y F B mx+c .


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The error or residual! may be rewritten as

y B y* + e

or e B y y*

y

y*

x

y

$igure 5e ;inear regression and error definition

The Gbest@ straight line would be the one that minimi"e the total

error. %everal criteria may be used. -owever, a more practical

criterion for least0squares approach is to minimi"e the sum of the

squares of the residuals Sr !, that is

minimi"e %x B∑

=

n

i

ie1

2

( )∑=

−−n

i

iii cmx y

1

2B


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Thus, we need to determine the value of m and c that fulfills the equation. This can

be achieved by differentiating with respect to each coefficient and set the

derivatives equal to "ero. The differentiations are not presented here. The resulting

expressions are:

nc C m ∑ xi B ∑ yi

c ∑ xi C m ∑ xi! B ∑ xi yi

n x

x x

c

m

y

x y

i

i i

i

i i

∑

∑ ∑

∑

∑

=

2

In matrix

formBH

( )∑ ∑

∑ ∑ ∑

−

−

22

ii

iiii

x xn

y x y xn

n

xm y ii∑ ∑−

These equations are called the normal

e"uations. They can be solved using

(ramer@s rule for m and c to obtain

m B

( B


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Error uantification in ;inear 'egression

To determine whether the straight line is a good fit to the set of

data or not, one may need to compute the coefficient of correlationr in the following manner

( ) ( )∑ ∑∑ ∑

∑ ∑ ∑

−•−

−

2222iiii

iiii

y yn x xn

y x y xn

r =

Interpretation of r ,r B ± / : the straight line is a good fit to the data i#e all the data lie precisely on the line!

r B ? : the straight line is not a good fit to the data i#e data scattered randomly and do

not fit a straight line!

? J r J / : /??r 1 K of change in x is accounted for change in y .

$or example, when r B ?.8, then /???.8!2 B 63 K which may be interpreted as 63K of change in y is due to change in x .


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%tatistical 9nalysis

%cience of decision ma#ing in uncertain situation

[ ] ( )5.0

25.0

1 ,:2

),(tan:,

−=

−== ∑∑

∑=

n

x x populationentireSD

n

x x sSDdeviationdard s

n

x

xmeanii

n

i

i

σ

( )

( )

1

,var :

1

2

2

222

−

−=

−

−

= ∑∑∑n

x x siance

nn

x xn

si

ii$or small samples:


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Example : $it a least square line to following data and find trend value

There are 5 data points BH n B 5. (onstructing the table for various terms as

follows.

x i y i x i 2 x i y i y i 2

Σ

(ombining relevant equations, m$ c , and r are obtained

X 1 2 3 4 5 2 3 5 6 7


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Ch4 Data Acquisition

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