Lecture # 2: Measurement Uncertaintieshuhui/teaching/2017-01Sx/AerE344/...Dally & Riley, “Experimental Stress Analysis,” Part V Ch 21 Labs begin! • Labs begin this week • Please

Copyright © by Dr. Hui Hu @ Iowa State University. All Rights Reserved!

Dr. Hui Hu

Dr. Rye Waldman

Department of Aerospace Engineering

Iowa State University

Ames, Iowa 50011, U.S.A

Lecture # 2: Measurement Uncertainties

AerE 344 - Lecture Notes


Dimensional analysis example

What determines the period of a

pendulum?

Parameters (5):

• m – [M]

• l – [L]

• g – [L T-2]

• t – [T]

• θ – [-]

Dimensions: M, L, T (3)

Number of Pi terms: 5-3 = 2

Choose independent variables: m, l, g

Create Pi terms from remaining variables

t and θ

[-] = t * ma lb gc = [T] * [M]a * [L]b * [L T-2]c

T: 0 = 1 – 2c → c = 1/2

M: 0 = a → a = 0

L: 0 = b + c → b = -1/2

[-] = θ = [-]

Therefore Pi groups are θ and

m

l

g θ

gt

l

Therefore, the only possible explanation for the

period of the pendulum must have a form:

Compare to the actual period of a pendulum:

The mass does not affect the period, ruled out

by purely dimensional arguments!

( ) ( )g l

t f t fl g

2 , 1l

tg

t


Sources/ Further reading: Tropea, Yarin, & Foss, “Springer Handbook of Experimental Fluid Mechanics,” Part D Ch 23

Dally & Riley, “Experimental Stress Analysis,” Part V Ch 21

Labs begin!

• Labs begin this week

• Please do the pre-lab assignment prior to your lab (available on the website)

• Please be at your lab on time

• Lab reports are due the Friday the week after the lab

• Office hours

• Tuesday 3–4

• Thursday 3–4


Calibration

• Calibration: A calibration applies a known input value to a measurement

system for the purpose of observing the system output value. It establishes the

relationship between the input and output values.

• The know value used for the calibration is called standard.

0

200

400

600

800

1000

1200

1400

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Experimental datacurve fitting

y=a+bx max dev:30.4, r2=0.998

a=13.2, b=240

Voltage (V)

Pre

ssu

re (

Pa

)


Example


Instrument Resolution

• Instrument Resolution represents the smallest increment in the measured value that can be discerned by using the instrument. In terms of a measurement system, it is quantified by the smallest scale increase of least count.

• Dynamic Range is the ratio of the extent of values that can be measured with the instrument: DR = Full scale / Resolution

Often given as number of “counts” “bits”

or “digits”

http://en.wikipedia.org/wiki/File:Super-resolution_example_closeup.png


Measurement Uncertainties

• “Accuracy” is generally used to indicate the relative closeness of agreement

between an experimentally-determined value of a quantity and its true value.

• “Error” is the difference between the experimentally-determined value and its

true value; therefore, as error decreases, accuracy is said to increase.

• Since the true value is not known, it is necessary to estimate error, and that

estimate is called an uncertainty, U.

• Uncertainty estimates are made at some confidence level—e.g., a 95%

confidence estimate, means that the true value of the quantity is expected to be

within the ±U interval about the experimentally-determined value 95 times out of

100.

truemtruemeasurederror AAEAAA

Which case is the more accurate measurement?

10 / , Measurement error 1 /

100 / , Measurement error 5 /

t

t

V m s V m s

V m s V m s

true

error

relativeA

AE



• Total error, U, can be considered to be composed of two components:

– a random (precision) component,

– a systematic (bias) component,

– We usually don’t know these exactly, so we estimate them with P and B,

respectively.

• Precision Error: Random error

– e.g., Normal Distribution or Gaussian Distribution

• Bias Error: Fixed error, system error

– Constant throughout the experiment

– Can be positive or negative

222 PBU

X

True value

X=100

Bias error precision

error

measured value

X=101



• Precise but biased

• Unbiased but Imprecise

• Biased and Imprecise

• Precise and Unbiased

222 PBE

Qualification of measurement error:


Measurement Results

0

200

400

600

800

1000

1200

1400

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Experimental datacurve fitting

y=a+bx max dev:30.4, r2=0.998

a=13.2, b=240

Voltage (V)

Pre

ssu

re (

Pa

)

Bias error

Precision error:

R.M.S of your data


Repeatability and Reproducibility

• Repeatability is the variability of the measurements

obtained by one person while measuring the same

item repeatedly. This is also known as the inherent

precision of the measurement equipment.

Consider the probability density functions shown in

Figure 1. The density functions were constructed

from measurements of the thickness of a piece of

metal with Gage A and Gage B. The density

functions demonstrate that Gage B is more

repeatable than Gage A.

Figure 1. Repeatability


Repeatability and Reproducibility

Repeatability Precision Error

Reproducibility Both Bias and Precision Errors

• Reproducibility is the variability of the measurement

system caused by differences in operator behavior.

Mathematically, it is the variability of the average

values obtained by several operators while

measuring the same item.

Figure 2 displays the probability density functions of

the measurements for three operators. The

variability of the individual operators are the same,

but because each operator has a different bias, the

total variability of the measurement system is higher

when three operators are used than when one

operator is used.

Figure 2. Reproducibility

Figure 1. Repeatability



• We almost always are dealing with a data reduction equation to get to

our final results.

– In this case, we must not only deal with uncertainty in the

measured values but uncertainty in the final results.

• A general form looks like this:

– R is the result determined from J independent variables.

“Propagation of errors”

JXXXXRR ,,,, 321

1 2

1 2

J

J

R R RdR dX dX dX

X X X


Example

pppV

BernoulliVpp

statictotal

statictotal

2)(2

)(,2

1 2

222

RRR PBU

Uncertainty in velocity V:

J

i

i

i

R

J

i

i

i

R PX

RPB

X

RB

1

2

2

1

2

2 ;

M

j

ii jBB

1

2

For a large number

of samples (N>10) ii SP 2

N

kkii

N

k

ikii XN

XXXN

S1

21

1

2 1;

1

1


Example

pppV

BernoulliVpp

statictotal

statictotal

2)(2

)(,2

1 2

For a large number

of samples (N>10)

N

kkii

N

k

ikii XN

XXXN

S1

21

1

2 1;

1

1

Error bars:

Precision error

Bias error


Lab #1: Flow visualization by using smoke wind tunnel

• Path line

• Streak lines

• Streamline


Streamlines (experiment)

Lab #1: Flow visualization by using smoke wind tunnel

Documents

Lecture # 2: Measurement Uncertaintieshuhui/teaching/2017-01Sx/AerE344/...Dally & Riley, “Experimental Stress Analysis,” Part V Ch 21 Labs begin! • Labs begin this week • Please