Engineering Measurements

1Engineering Measurements

Indrawanto

Mechanical Engineering Department

Faculty of Mechanical and Aerospace Engineering

ITB

2013

Textbook

Indrawanto, FTMD-ITB, Engineering Measurements, 2013

Textbook


Purpose of Measurement

The purpose of a measurement system is to present an observer with a numerical value corresponding to the variable being measured


2General structure of measurement

system

The elements of measurement system can be defined as

Sensing element: device which detects and responds to measurand

Signal conditioning element: amplify, filter, integrate, differentiate, convert freq. to voltage, etc.

Signal processing element

Data presentation element

Measurements Direct/Indirect comparison (rulers, balance scale, interferometer) Calibrated system (odometer, spring scale, pressure gage)


Example of Measurement system

Tire Pressure Gage Bourdon Type Pressure Gage


Computers Readout SystemsSequential Sampling Data Acquisition

Simultaneous Sampling Data Acquisition

MUX = multiplexer (switch)S/H = sample & hold (hold voltage while ADC reads)ADC = analog to digital converter


An Example of a Measurement system


3Block diagram symbols


Static characteristics of

measurement system elements


Systematic characteristics

Accuracy

Precision

Ideal straight line

Non-linearity

Sensitivity

Environmental effects

Hysteresis

Resolution

Wear and ageing

Error bands


Static characteristics of

Measurements Accuracy: difference between measured and true values

(error); typically specified by a maximum value.

error = = xmeas - xexact

Precision: difference between measured values during

repeated measurements of the same quantity.

Types of error

Accuracy &Precision

Precision(can be calibrated)

Accuracy(can be averaged)

Neither


4Ideal straight line

( )MINMINMAX

MAXMINMIN IIII

OOOO

=

aIKO +=IDEAL

MINMIN

MINMAX

MINMAX

KIOaIIOOK

==

==

intercept line-straight ideal

slope line-straight ideal

Ideal straight line


Definition of Non-Linearity

( ) ( ) ( ) ( ) ( )

( ) =

=

=++++++=

=

++=+=

mq

q

qq

m

m

qq IaIaIaIaIaaIO

OON

INaKIIOaKIIOIN

0

2210

MINMAX

%100

f.s.d of procentage as NLMax

or

LL


Sensitivity

Rate of change of O with respect to I

INK

IO

dd

dd

+=

Sensitivity: change of an instruments output per unit change in the measured quantity


Environmental effects

( ) IIMM IKIIKINaKIO ++++=Modifying Interfering

Thermocouple

sensitivity.


5Hysteresis

( ) ( ) ( )%100

..

=

=

MINMAX

II

OOHdsfprocashysteresisMax

IOIOIHHysteresis


Example: Histerisys, Backlash in Gears


Resolution

Resolution is defined as the largest change in I that

occurs without any corresponding change in O.

Resolution expressed as a percentage of f.s.d.

%100

MINMAX

R

III

Resolution and potentiometer example.


Wear and Ageing

Measurement elements change slowly but

systematically throughout their life.

One example is the stiffness of a spring k(t)

decreasing slowly with time due to wear, i.e.

k(t) = k0 bt


6Error bands

Nonlinearity, hysteresis and resolution effects in many sensors and transducers are small that is difficult and not worthwhile to exactly quantify each individual effect. In these cases the manufacturer defines the performance of the element in terms of error band.


Error bands (con.)

Error bands and rectangular probability density function.

Probability density function.Indrawanto, FTMD-ITB,

Engineering Measurements, 2013

Generalized model of a system

element


Generalized model of a system element

(strain gauge)


7Generalized model of a system element

(thermocouple)


Generalized model of a system element

(accelerometer)


Statistical characteristics

Probability & Statistics provides a way to deal with uncertainty

Bias error: systematic errors that can be removed by calibrationPrecision error: random error that isnot directly controllable (by definition), could be minimized by careful design and statistical analysis

Bias:calibrationconsistent human errorloading error

Precision:disturbancesnoisevariable conditions


Statistical characteristics (con.)

+

+

=2

22

22

2

III

MM

IO IO

IO

IO

II

MM

IIOI

IOI

IOO

+

+

=

Small deviation from the mean value

Standard deviation of output for a single element

Normal probabilitydensity function


8Statistical characteristics (con.)

( ) IIMM IKIIKINaIKO ++++=

( ) ( )

= 2

2

2exp

21

pi

OOOpo

Mean value of output for a single element

the corresponding probability density function



Statistical variations amongst a batch of similar

elements tolerance

( ) ( )

= 2

200

000

2exp

21

RR

RRRppi

The Gaussian probability density function

= mean value of distribution0R





( ) IIMM IKIIKINaIKO ++++=

+

+

+

+

+

= L22

22

22

22

22

aIII

IM

Ia

OKO

IO

IO

IOO

KIM

Mean value of output for a batch of elements

Standard deviation of output for a batch of elements


9Model for chromelalumel thermocouple.


Model for millivolt to current temperature transmitter.


Calibration


Calibration


10

Calibration


Measurement error of a system of

ideal elements

Ideal output

O = On = K1 K2 K3 Ki KnI

E = (K1K2K3Kn -1)


Example


General calculation of system


11

General calculation of system


Example




12

Model for recorder


Summary of calculation


Error band


Error Reduction Technique


13





Closed-loop


Dynamic characteristics of

measurement systems

Dynamic behavior is described by the heat balance equation:

Rate of heat inflow rate of heat outflow = rate of change sensor heat content


14

( )( )[ ]

( )

F

F

F

TTdt

TdUAMC

dtTdMCTTAU

TTdtdMC

TTAUW

=+

=

=

=

i.e.

0 content heat sensor of increase of rate

watts

First order linear differential equationIndrawanto, FTMD-ITB,


==

secs

WJ

mCmWCkgJkg

21-02-

-10-1

The quantity MC/UA has the dimensions of time and is refered to as the time constant tttt


( ) ( ) transformLaplace of Definition 0

dttfesf st

=

( ) ( )[ ] ( ) ( )sTsTTsTs F=+ 0( ) ( ) ( )

( ) ( ) ( )sTsT

sTsTsTs

F

F

=+

=+

1si.e.

Using Laplace table

At initial conditions equal zero

eq. diff.order first Linear FTTdtTd =+


Laplace transform


15

Transfer function


Analogous First Order Elements



RCVVt

V

VVt

VRC

t

VCt

qiCVq

iRVV

EE

in

==+

=+

===

=

,dd

i.e.dd

dd

dd

current , Charge

IN

IN



kFF

t

F

FFt

Fk

kkF

x

t

xFF

MM

==+

=+

==

==

,

dddd

stiffness springNm,nt displaceme

constant dampingNsmdd

IN

IN

1

1IN


16


kCRCR

k

CCCRRCg

ACRRCRgRA

MCCUA

RCRUAMC

MMMM

gggg

FFFFFF

FFF

ThThThThTh

1,; Mechanical

, Electrical

,; Fluidic

,

1; Thermal

M ====

===

====

====


Mass-spring-damper model for elastic

sensor

FkxxxmxmxkxF

=++

=

=

&&&

&&&

onaccelerati mass forceresultant


sec/rad m

kn = km2

=

Fk

xt

x

t

x

nn

=++ 1d

d2d

d12

2

2

Undamped natural frequency

Damping ratio

( ) ( )xxxx

xxxFFF&&&& ==

==,

0,0

Fk

xt

x

ktx

km =++ 1

dd

dd

2

2

Linear second order differential equation


( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )sFk

sxxsxsxxssxsnn

=++ 102001 22

&

( ) ( )sFk

sxssnn

=

++

1121 22

( )( ) ( )sGksFsx 1

=

( )121

12

2 ++=

ss

sG

nn

Transfer function for a second-order element


17

Series R-L-C circuit

= +

+

where =

( = charge on the capacitance)

thus 2

2+

+

1

=


Identification of the dynamics of an element

Step response of first order elements

( ) ( ) ( ) ( )sssfsGsf i +== 11

0

Expressing in partial fractions

( ) ( ) ( ) sB

s

Ass

sf ++

=

+=

111

0

B=1, A=-t

( ) ( ) ( ) /111

11

0+

=

+=

sssssf


Using Laplaces table( )

=

tsf exp10

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1


Second order elements

( )121

12

2 ++=

ss

sG

nn

( )sss

sf

nn

++

=

1211

22

0

( )s

C

ss

BAssf

nn

+

++

+=

121 220

Output signal due to a unit step input

Expressing inPartial functions


18

( ) ( )

( )( ) ( )

( )( ) ( ) ( ) ( )222222

222

220

111

121

221

++

++

+=

++

+=

++

+=

nn

n

nn

n

nn

n

nn

n

ss

s

s

s

s

s

ss

s

ssf

( ) ( )2011

n

n

n ssssf

+

+=

( ) ( )tetf ntn += 110


( ) ( ) ( ) ( )

+= ttetf nntn 222

0 1sin1

1cos1

( ) ( ) ( ) ( )

+= ttetf nntn 1sinh1

1cosh1 22

20


Sinusoidal response of first order elements

( ) ( ) ( )220 11

++=

s

Is

sf

( ) ( ) 220 1 ++

++

=

s

cBss

Asf

( ) ( ) ( )2222222

1,

1,

1

+=

+

=

+=

ICIBIA


( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )

+

+

++

++=

+

+

++

++

++=

+

+

++

++=

222222

2

22

2222

2222

2

222222

2

0

sincos11

11

111

111

1

111

1

s

sIs

I

s

sI

s

I

s

sIs

Isf


19

( ) ( )2222 1sin,11

cos

+

=

+=

( ) ( ) ( )4444 34444 21

44 344 21

termSinusoidal

22

termTransient

/220 sin11

++

++

= t

Ie

Isf t


-40

-30

-20

-10

0

M

a

g

n

i

t

u

d

e

(

d

B

)

10-2

10-1

100

101

102

-90

-45

0

P

h

a

s

e

(

d

e

g

)

Bode Diagram

Frequency (rad/sec)


-40

-30

-20

-10

0

10

20

M

a

g

n

i

t

u

d

e

(

d

B

)

10-1 100-180

-135

-90

-45

0

P

h

a

s

e

(

d

e

g

)

Bode Diagram

Frequency (rad/sec)

( )121

12

2 ++=

ss

sG

nn

( )( ) ( ) 121

12

2 ++=

jj

jG

nn


( )

=

2

22

2

2

2

41

1

nn

jG

( )

=

221

/1/2

tanargn

njG


20

Dynamic errors in measurement

systems

( )sOi( ) ( )sIsO 21 =

Complete measurement system with dynamics

K2G2(s) KiGi(s) KnGn(s)K1G1(s)

1 2 3 4Input i.e. true signal

Output i.e. measured signal

( ) ( )sIsI 1= ( )sI i( )sO2 ( )sIn ( )sO


( ) ( ) ( )tItOtE =

( )( ) ( ) ( ) ( ) ( ) ( )sGsGsGsGsGsI

sOni LL21==

Transfer function for complete measurement system


Simple measurement system with

dynamics


( ) ( ) ( )}2001{20 200101.0 4 tEeBeAetutT tttM ++=

( ) ( ) ( ) 2420011

11011

1011120

+

++=

sss

s

( ) ( ) ( )

+

+

++= 24 200101.0

120s

DCss

Bs

As

( ) ( )tTtTtE TM = )(( )}2001{20 200101.0 4 tEeBeAe ttt ++=


21

( ) ( ) ( ) ( ) +== tIjwGtOtItI sinsin

( ) ( )( ) ( )( )2524 105.2101101101 1 jjjjjG ++++=( )

( )( ) ( ) 10.0]}10105.21[1011001{1

42581

+++=

=jG

( ) ( ) ( ) =

8510tan10tan10tan0)(arg 21411jG

( ) ( ){ }tttE sin85sin1.020 =Indrawanto, FTMD-ITB,


Periodic Functions

A function f(t) is said to be periodic it is defined for all real t and if there is some

positive number p such that

( ) ( )tfptf =+For any integer n

( ) ( )tfnptf =+ for all tIf f(t) and g(t) have period p, then

( ) ( ) ( )tbgtafth += (a,b constant)


Periodic Signals

A periodic signal may have:

( )L

L

+++

+++=

tbtbtataatf

0201

02010

2sinsin2coscos

This may be written as

( )

=

=

++=1 2

000 sincosn n

nn tnbtnaatf

where

22

Periodic Signals Contd

( )

dttmtnb

dttmtna

dttmadttmtf

T nn

T nn

T T

01

0

01

0

000

cossin

coscos

coscos

0

0

0 0

+

+

=

=

=

Multiplying both sides by cos mw0t and integrating over a period of f(t),

we obtain



The first term integrates to zero. We multiply each term of the seriesin parentheses by cos w0t and integrate term by term to obtain

( )

=

=

+

=

100

01

00

0

0 0

cossin

coscoscos

n Tn

T n Tn

dttmtnb

dttmtnadttmtf

Three possible cases occur


The resulting integrals are

nmdttntmI

nmTnm

dttntmI

nmTnm

dttntmI

T

T

T

, all,0cossin

0,2/,0

coscos

0,2/,0

sinsin

0

0

0

003

0002

0001

==

=

==

=

==

and



( )

= 2cos

00

0

Tadttmtf m

T

( ) =0

0,cos2 00 T

m mdttmtfTa

( )=0

00

sin2

Tm dttmtfTb

For n=m, the integral gives T0/2, which yields

Multiplying both sides by T0/2

In similar manner it can be shown that


23

Example

A square wave signal f(t) is defined as

And periodically extended outside this interval. The average value is zero, so a0=0, therefore ams are

( )

24

Periodic Signals

( )( )( ) ( ) Ttfttf

Ta

ttntfT

b

ttntfT

a

T

T

T

Tn

T

Tn

over of valueaveraged1

dsin2

dcos2

2/

2/0

2/

2/ 1

2/

2/ 1

==

=

=

+

+

+

( ) =

=

=

=

++=n

n

n

n

nn tnbtnaatf1 1

110 sincos Fourier series for periodic signal



( )

( ) ( ) ( )

( ) ( ) ( ){ }

=

=

=

=

=

=

+=

+=

=

n

n

nn

n

n

nn

n

n

n

tntnjnGItE

tnjnGItO

tnItI

1111

111

11

sinsin

sin

sin


Damped harmonic oscillator

Excitation

Response

0 = 02 =


Response of damped harmonic oscillator to

sawtooth excitation

Excitation

Response

02 = 03 =0 =


25

Response of damped harmonic oscillator to

sawtooth excitation

20 =

0 =2

3 0 =

Excitation

Response


( ) ( ) ( ) ( ) 014.07,020.05,033.03,100.0 jGjGjGjG

( ) ( ) ( )[( ) ( )]00

00

937sin002.0925sin004.0

903sin011.085sin100.080

++

+=

tt

tttTM pi

( )

++++= LtttttTT 7sin7

15sin513sin

31

sin80pi

( ) ( )( ) ( ) 00

00

937arg,925arg903arg,85arg

jGjGjGjG


Solution of differential equation using Laplace Transform

Partial-fraction Expansion: Distinct Real Roots

( ) ( )( )( )( )3142

++

++=

sss

sssY

( )31

321

++

++=

s

Cs

Cs

CsY

( )( )( )( )( )( )

( )( )( )

( ) 61

142

23

342

38

3142

33

12

01

=

+

++=

=

+

++=

=

++

++=

=

=

=

s

s

s

ss

ssC

ss

ssC

ss

ssC

( ) ( ) ( ) ( )tetetty tt 1611

231

38 3

=


Solution of differential equation using Laplace

Transform

Partial-fraction Expansion: Distinct Complex Roots

( ) ( )11

2 ++=

ssssF

( )12321

++

++=

ss

CsCs

CsF

( ) 101 == =sssFC( ) ( ) 11 322 =++++ sCsCss

( ) ( ) 4322121

211

++

++=

s

s

ssF

( ) ( )ttetetf tt 1sin3

1cos1 43

2/432/

=


26

Solution of differential equation using Laplace

Transform

Partial-fraction Expansion: Distinct Repeated Roots

( ) ( )( )2213++

+=

ss

ssF

( ) ( )2321

221 ++

++

+=

s

Cs

Cs

CsF

( ) ( ) ( )( ) ( )

( ) ( ) 1132

22

2231

22

23

2

22

1211

=

+

+=+=

=+=

=

+

+=+=

=

=

=

=

=

ss

s

s

s

s

ssFsC

sFsdsdC

s

ssFsC

( ) ( ) ( )tteeetf ttt 122 22 =Indrawanto, FTMD-ITB,


Example 2

A pressure sensor has the following transfer function

Input signal Pin to the sensor can be represented as follow

Show the sensor output due to the second harmonic of the input signal

( ) ( )( ) 121+

==

ssPsP

sGin

out

0 1-1

10

T second

P barin


Signals

Deterministic signals: a deterministic signal is

one whose value at any future time can be

exactly predicted

Random signals: a random signal is one whose

value at any future time can not be exactly

predicted


Deterministic and Random Signals


27

Statistical Representation of Random Signals

Sampling of a random signal

T0 is sufficiently long, i.e. N is sufficiently largeThe signal is stationary, i.e. long term statistical quantities do not change with time



Contd

( ) ttyT

yT

d1 00

0=

=

=

=

Ni

iiyN

y1

1

( )[ ] tytyT

T

d10

0

2

0

2 = ( )

=

=

=

Ni

ii yyN 1

22 1

Mean for continuous signal

Mean for sampled signal

Standard deviation for continuous signal

Standard deviation for sampled signal



Contd

0=y

=0

02

0rms d

1 Tty

Ty

=

=

=

Ni

iiyT

y1

2

0rms

1

For special case

The r.m.s. value is

or


Probability Density Function p(y)

Probability Density Function is function of signal value y and is a measure of the probability that the signal will have a certain range of values

Sampling of a random signalIndrawanto, FTMD-ITB,


28

Probability Density Function p(y)

mjNn

jP

j

j

,,1,

samples ofnumber totalsectionth in the occurs sample timesofnumber

K==

=The probability


( )jjj

nnnN

PPPC

+++=

+++=

L

L

21

21

1

( )( )NnnnN

N

nnnN

C

m

mm

=+++==

+++=

L

L

21

21

since11

1

The cumulative probability

The final value of Cj when j=m is:


( ) jy CyP 0lim=Cumulative probability distribution function

( )yPyp

dd

=

Probability density function


( ) yypPP yyy ==+,( )= 2

121

d,

y

yyyyypP

( ) ( )

= 2

2

2exp

21

pi

yyyp

Probability for signal that lie between y and y + y is

Probability for signal that lie between y1 and y2 is

The Gaussian probability density function is:


29

Power Spectral Density

Signal power is a stationary quantity which can be used to quantify random signals.

( ) =

=

=

=

++=n

n

n

n

nn tnbtnaaty1 1

110 sincos

The Fourier series for a signal with period T0with yaT mean,/2 001 == pi Indrawanto, FTMD-ITB,


( )

( )

=

=

=

=

=

=

=

=

==

==

Ni

ii

Ni

iin

Ni

ii

Ni

iin

Ni

nyN

TinyN

b

Ni

nyN

TinyN

a

111

111

2sin2sin2

2cos2cos2

pi

pi

Fourier series coefficients for sampled signal


Power Spectrum

( )2221 nnn baw += nn wwwwW ++++= L210Power due to nth harmonic Cumulative power


Power Spectrum Contd

( ) nWW 01lim= ( ) ddW

w =

Power spectral density in watts sec rad-1 or watts/Hz

Cumulative power function


30

Autocorrelation function

( ) ( ) ( ) =

0

0 00

d1limT

Tyyttyty

TR

Autocorrelation function Ryy() of the signal is the relation between Ryyand time delay .

A Simple Correlator

Autocorrelation function of a continuous signal


Autocorrelation function Contd

( ) ( )[ ]

12

0 10

2

00

1

2

cos

d22cos1lim2

d1limcos2

0

0

0

0

b

ttT

bt

TbR

T

T

T

Tyy

=

+=

( ) ( ) += tbty 1sin If( ) ( ) ( )[ ]

( )[ ]{ } ttT

b

tttT

bR

T

T

T

Tyy

d22coscos1lim

dsinsin1lim

0

0

0

0

0 1121

0

2

0 110

2

+=

++=

Thus the autocorrelation function of a sinusoidal signal is cosine function of the same frequency, but the phase information in the sine wave is lost



K,2,1,0, == mTm( )

=

=

=

Ni

imiiNyy

yyN

TmR1

1lim



( ) tbtbtbty 131211 3sin2sinsin ++=A periodic signal which is the sum of three harmonics

The autocorrelation function

( ) 133

22

21 3cos

22cos

2cos

2bbbRyy ++=


31


( ) ( )( ) ( )

pi

dcos2

dcos

0

0

=

=

yy

yy

R

R

( )

C

yy

A

AARC

C

sin

sindcos0

0

=

==

The power spectrum can be obtained from the autocorrelation function by Fourier analysis. Similarly the autocorrelation function can be obtained from the power spectrum by adding the harmonics together. For random signals, Ryy() and () are related by the Fourier transform:

For a signal with () constant up to C and zero for higher frequencies then


Summary

In order to specify a random signal we need to know:

deviation standard and meanfunction density yprobabilit

or

either

function ncorrelatio autodensity spectralpower

or

either

To specify amplitude behavior

To specify frequency/time behavior

or

( ) ( ) dd1lim000

2

0

0

0

==

T

Tyyty

TR

( ) TOT2RMS0 WyRyy ==The autocorrelation coefficient at zero time delay


Engineering Measurements

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