AE2610 Introduction to Experimental Methods in Aerospace

AE2610Introduction to

Experimental Methods in Aerospace

Measurement Systems & Plotting

C.V. Di Leo

1

Sensing/Measurement Systems

2

• Measurements• Absolute• Relative/calibrated

• Sensing/recording• Human based• Mechanical• Electrical

(Digital) Data Acquisition Systems

3

• Detector: device which detects and responds to measurand.

• Transducer: converts measurand to an analog more easily measured (e.g. displacement to voltage).

• Signal conditioner: amplify, filter, integrate, differentiate, etc.

• Recording: microprocessor-based based device nowadays (includes dedicated systems, computers, smart phones, ….)

Transducer/Sensor Characterization• Sensitivity

• Change in a measurement device’s output for a unit change in the measured (input) quantity, e.g., volts/Torr for the Baratron

• Resolution• Smallest increment of change

in a system or property that a measurement device can reliably capture

• Dynamic Range• Maximum output of a

measurement device divided by its resolution (or minimum measureable signal)

4

Resolution = 50 psiDyn. Range= 5000/50

= 100

Sensitivity= 5000 psi/270°= 18.5 psi/degree

0° 270°

Computer Data Acquisition

5

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

Volt

t

Chan #1

Chan #2

skew

sample

Readings are not recorded simultaneously

S/H voltages

digital words

to compute

r MUX ADC Buffer/Board Memory

Computer/Main Memory

Master Clock/

Controller

system bus

Acquisition Board

SequentialSamplingDataAcquisition

Volt

t

Chan #1

Chan #2

No skew

sample

Readings arerecorded simultaneously

S/Hvoltages

digitalwords

tocompute

rMUX ADC Buffer/BoardMemory

Computer/MainMemory

MasterClock/

Controller

system bus

AcquisitionBoardS/H

S/HS/HS/HS/HS/HS/H

SimultaneousSamplingDataAcquisition

Computer Data Acquisition• The (multiple) S/H -> Mux -> S/H -> ADC architecture is based on

the fact that historically S/H units were much cheaper than ADCs.• The trade-off is that the sample acquisition must be shared by the

number of channels!• However, steady decrease in cost of ADC units has lead to newer

architectures that are really parallel:• multiple (S/H -> ADC) units in parallel (no Mux)

6

Graphing

7

Graphing• Graphs are one of the most effective ways to

communicate experimental results• However, graphs are often constructed poorly.

8

• Graphs convey- Trends- Outliers- Uncertainty- …

Time Pressure

10:10 1009.0

11:30 984.2

1:00 999.8

2:15 989.9

3:40 977.1

4:40 981.2

5:40 990.0

clear legend(also explains what the

line denotes)

symbols usedfor experimental data

legible and meaningfulx and y labels

Clear Graphing

9

Spring 2001 AE 3145 Measurements Lab 2

Make it Clear to Reader...

Good:• y=values on log scale

• x=every 20 years

• Circle symbols

• Linear trend...



• Circle symbols

• Linear trend...

Poor:• y=log()

• x=too many values

• No data symbols

• What does line mean?

Poor:• y=log()


• No data symbols



Make it Clear to Reader...



• Circle symbols

• Linear trend...



• Circle symbols

• Linear trend...

Poor:• y=log()


• No data symbols


Poor:• y=log()


• No data symbols


Poor Graph• y = (value) • x – too many values,

porly spaced• No date symbols• Solid line not defined

Good Graph• y – log scale• x – evenly spaced• Data circle symbols• Trend line explained

Basic Types of Cartesian Plots

10

Linear• Both axes are linear• Symbols for data and lines

for trend/fits

Basic Types of Cartesian Plots

11

• For Magnesium, Fig. S2, insufficient data for pure magnesium was available in the literature. Hence,we considered experimental results for the alloys AZ31, AZ61, and AZ91 as shown in Fig. S2. In ouranalysis, we included only the experimental data for the Magnesium alloy AZ91.

• All of the experimental data extracted for Chromium, see Fig. S3, was subsequently used in the mainbody of the publication.

• When selecting experimental results for Iron, care was taken to extract experimental data for materialswith low carbon content (on the order of 0.02%C weight per cent or below), and to ensure that thematerial retains a BCC structure. Further, whenever a yield points was experimentally measured, andboth upper and lower yield stresses where measured, the lower yield point is reported here.

Copper Aluminum Nickel

d (µm)10-2 100 102

σ(M

Pa)

0

200

400

600

800

1000

all dataconsidered

d (µm)10-1 100 101 102 103

σ(M

Pa)

0

50

100

150

200

250

300

350

400

d (µm)10-2 100 102

σ(M

Pa)

0

500

1000

1500

2000

2500

d (µm)10-2 100 102

σ(M

Pa)

0

200

400

600

800

1000

data includedin analysis

d (µm)10-1 100 101 102 103

σ(M

Pa)

0

50

100

150

200

250

300

350

400

d (µm)10-2 100 102

σ(M

Pa)

0

500

1000

1500

2000

2500

Gourdin & Lassila (1991)Youngdahl (1997), HardnessFeltham and Meakin (1957)Lu et al. (2009)Shih et al. (2001)Flinn et al. (2001)Chokshi et al. (1989), VHHayashi and Etoh (1989), VHVinogradov et al. (2006)Valiev et al. (1994)Thompson and Backofen (1971a)Thompson and Backofen (1971b)Li et al. (2008)Sanders et al. (1997), VH

Tsuji et al. (2002)Yu et al. (2005)Hansen (1977), 99.5%Fujita and Tabata (1973), at 293KHorita et al. (2000), 1100Horita et al. (2000), 3004Yu et al. (2005) at 77KHansen (1977), 99.999%Fujita and Tabata (1973), at 373KFujita and Tabata (1973), at 200KFujita and Tabata (1973), at 77KCarreker and Hibbard (1955)

Keller and Hug (2008)Floreen & Westbrook (1969), (<1ppm S)Ebrahimi et al. (1999)Xiao et al. (2001)Schuh et al. (2002), HardnessHughes et al. (1986), HardnessThompson (1975), RolledThompson (1975), ElectroplatedFloreen & Westbrook (1969), (22ppm S)Krasilnikov et al. (2005)

Figure S1: Experimental yield stress vs. grain size data for FCC materials extracted from the literature. Thetop row shows all of the experimental data extracted, where square symbols show the data subsequently usedfor analysis in the main body of this publication, and all other symbols are excluded. The data subsequentlyused in the main body is also shown on its own in the second row of figures. The legends in the third rowshow the corresponding references from where the experimental data was extracted.

2

• For Magnesium, Fig. S2, insufficient data for pure magnesium was available in the literature. Hence,we considered experimental results for the alloys AZ31, AZ61, and AZ91 as shown in Fig. S2. In ouranalysis, we included only the experimental data for the Magnesium alloy AZ91.

• All of the experimental data extracted for Chromium, see Fig. S3, was subsequently used in the mainbody of the publication.

• When selecting experimental results for Iron, care was taken to extract experimental data for materialswith low carbon content (on the order of 0.02%C weight per cent or below), and to ensure that thematerial retains a BCC structure. Further, whenever a yield points was experimentally measured, andboth upper and lower yield stresses where measured, the lower yield point is reported here.

Copper Aluminum Nickel

d (µm)10-2 100 102

σ(M

Pa)

0

200

400

600

800

1000

all dataconsidered

d (µm)10-1 100 101 102 103

σ(M

Pa)

0

50

100

150

200

250

300

350

400

d (µm)10-2 100 102

σ(M

Pa)

0

500

1000

1500

2000

2500

d (µm)10-2 100 102

σ(M

Pa)

0

200

400

600

800

1000

data includedin analysis

d (µm)10-1 100 101 102 103

σ(M

Pa)

0

50

100

150

200

250

300

350

400

d (µm)10-2 100 102

σ(M

Pa)

0

500

1000

1500

2000

2500

Gourdin & Lassila (1991)Youngdahl (1997), HardnessFeltham and Meakin (1957)Lu et al. (2009)Shih et al. (2001)Flinn et al. (2001)Chokshi et al. (1989), VHHayashi and Etoh (1989), VHVinogradov et al. (2006)Valiev et al. (1994)Thompson and Backofen (1971a)Thompson and Backofen (1971b)Li et al. (2008)Sanders et al. (1997), VH

Tsuji et al. (2002)Yu et al. (2005)Hansen (1977), 99.5%Fujita and Tabata (1973), at 293KHorita et al. (2000), 1100Horita et al. (2000), 3004Yu et al. (2005) at 77KHansen (1977), 99.999%Fujita and Tabata (1973), at 373KFujita and Tabata (1973), at 200KFujita and Tabata (1973), at 77KCarreker and Hibbard (1955)

Keller and Hug (2008)Floreen & Westbrook (1969), (<1ppm S)Ebrahimi et al. (1999)Xiao et al. (2001)Schuh et al. (2002), HardnessHughes et al. (1986), HardnessThompson (1975), RolledThompson (1975), ElectroplatedFloreen & Westbrook (1969), (22ppm S)Krasilnikov et al. (2005)

Figure S1: Experimental yield stress vs. grain size data for FCC materials extracted from the literature. Thetop row shows all of the experimental data extracted, where square symbols show the data subsequently usedfor analysis in the main body of this publication, and all other symbols are excluded. The data subsequentlyused in the main body is also shown on its own in the second row of figures. The legends in the third rowshow the corresponding references from where the experimental data was extracted.

2

Yie

ldStr

ess,

�y

(MPa)

Grain Size, d (µm)

σ0 ·G · b101 102 103 104

k

101

102

103

CuAlNiTiMgCrFek = γ(σ0Gb)1/2 with γ = 9.44

Figure 3: Plot of the Hall-Petch coefficient k vs. the quantity (σ0 ·G · b). Here, σ0 is fitted as the minimumyield stress experimentally measured for a given material, and k is subsequently fit using the Hall-Petchrelation σ = σ0+kd−1/2, also shown is our model prediction, eq. (7), with the average value of γ determinedearlier.

8

Hal

l-Pet

chC

oe�

cien

t,k

Log-Log• Both axes are log (could

use dB)• Symbols for data and lines

for trend/fits

Semi-Log• One axis is on a log scale• Symbols for data and lines for

trend/fits• Can also included error

bounds

Polar Chart• Useful for showing angular effects and variations along a

polar coordinate

12


Polar Chart

• Useful for showing angular effects

Three-Dimensional Plots• Show variations along three dimensions.

13

80

−200−100

0100

200

−200

0

200−200

−100

0

100

200

Iron

(a)

−500

0

500

−500

0

500−500

0

500

Tungsten

(b)

−200−100

0100

200

−200

0

200−200

−100

0

100

200

Niobium

(c)

Figure 6.5: Variation in the magnitudes of the effective modulus Ed with respect to the crystallographic axes of acubic crystal for (a) Fe, (b) W, and (c) Nb crystals. The (X,Y,Z)-scales in the figure are in GPa.

80

−200−100

0100

200

−200

0

200−200

−100

0

100

200

Iron

(a)

−500

0

500

−500

0

500−500

0

500

Tungsten

(b)

−200−100

0100

200

−200

0

200−200

−100

0

100

200

Niobium

(c)

Figure 6.5: Variation in the magnitudes of the effective modulus Ed with respect to the crystallographic axes of acubic crystal for (a) Fe, (b) W, and (c) Nb crystals. The (X,Y,Z)-scales in the figure are in GPa.

E�ective ModulusEd(GPa)

VariationinthemagnitudesoftheeffectivemodulusEd withrespecttothecrystallographicaxesofacubiccrystalfor

Semi-Log Graph: Revealing a Trend• Cool-down data from liquid sample shows exponential trend

when plotted using linear scales (left) and linear when plotted using log-y scale

• Experimental data shown in symbols• Theoretical results shown with curves• Legend used to define each symbol and curve

14


Semi-log Shows a Trend

• Cool-down data from liquid sample shows exponential trend when plotted using linear scales (left) and linear when plotted using log y scale.

• Experimental data shown with symbols.• Theoretical results shown with smooth curves.• Use Legend to define each curve or symbol used.

Changing Scales• Plotting nonlinear data using nonlinear axes can often

reveal underlying analytical form• Usually replotting is designed to create a linear curve

whose slope & intercept can be related to underlying parameters.

15


Changing Scales Can Show Structure

• Replotting nonlinear data using nonlinear axes can often reveal underlying analytical forms

• In this example, both asymptote and hyperbolic coefficient are revealed in intercept & slope of linear plot.

• Usually, replotting is designed to create a linear curve whose slope & intercept can be related to underlying parameters.

Nonlinear• y=1 + 2.5/x• Asymptotic behavior• Hyperbolic behavior


Nonlinear Axes• y=same; x →1/x• Slope = 2.5• Intercept = 1.0



Changing Scales Can Show Structure

• Replotting nonlinear data using nonlinear axes can often reveal underlying analytical forms

• In this example, both asymptote and hyperbolic coefficient are revealed in intercept & slope of linear plot.

• Usually, replotting is designed to create a linear curve whose slope & intercept can be related to underlying parameters.





Changing Scales• Plotting nonlinear data using nonlinear axes can often

reveal underlying analytical form• Usually replotting is designed to create a linear curve

whose slope & intercept can be related to underlying parameters.

16

Yie

ldStr

ess,

�y

(MPa)

Grain Size, d (µm)

Yie

ldStr

ess,

�y

(MPa)

d�1/2

Reveals that �y = Ad�1/2 + B

Plotting Uncertainty• Error bars should be plotted around the mean value

being reported to denote uncertainty. • Error bars can be both in the vertical and/or horizontal

axis.

17

Plotting Uncertainty• Error bars should be plotted around the mean value

being reported to denote uncertainty. • Error bars can be both in the vertical and/or horizontal

axis.

18

10

Copyright � 2002-2003,2008, 2016 by J. Seitzman and J. Craig. All rights reserved.errosr&uncertainty-26 AE2610

Plotting Uncertainty - Error Bars

• Can have error bars in vertical and/or horizontal coordinates

0

20000

40000

60000

80000

250 300 350 400

T (K)

p (P

a)

ArN2He

Uncertainty analysis

Linear Regression

19

20

Data reduction: Linear regression

How will we match the experimental data with our models?

In both the linear regime and in the strain hardening regime, weexpect linear, or piecewise linear behavior

� = P/A0

✏ = �/L0

proportional limit

yield stress

ultimate stress

strain hardeningnecking

�p

�y

�u

� = E ✏

rupture

ln ✏T

ln �T

ln �T = lnK + n ln ✏T

ln �T = lnE + ln ✏T

linear elastic regime

strain hardening regimelnK

n = slope

We’ll fit a straight line through our experimentally measured points

What is the best line that fits the data?

28 / 31

21

Linear regression continued

From our data, we have �i , Pi , and ✏i , measured throughout the testand initial values for L0 and A0

We can compute ✏�i from the elongation and �i from Pi and A0

What is the closest relationship � = E✏ between stress and strain inthe linear elastic regime?

In the linear elastic regime we’ll try and find a model that takes this form:

�i = E✏i + e

E is an estimate of the Young’s modulus and e is a biasSince not all points will lie on the line exactly, we’ll minimize theresidual:

ri = E✏i + e � �i

This leads to the unconstrained minimization problem

minE ,e

NX

i=1

r2i = minE ,e

NX

i=1

(E✏i + e � �i )2

29 / 31

22


Since the minimization is unconstrained, we can take derivatives withrespect to E and e

First, taking the derivative with respect to E gives:

NX

i=1

2✏i (E✏i + e � �i ) = 0

Next, taking the derivative with respect to e gives:

NX

i=1

2(E✏i + e � �i ) = 0

30 / 31

23


This leads to the following two equations in the unknowns E and e:"

NX

i=1

✏2i

#E +

"NX

i=1

✏i

#e =

NX

i=1

�i✏i

"NX

i=1

✏i

#E + Ne =

NX

i=1

�i

You can either compute this directly, or use tools in Excel or Matlabfor linear regression

To assess the fit, plot the residuals ri using the values E and e thatyou found

This procedure can be repeated for the linear relationship:

ln�T = lnK + n ln ✏T

31 / 31

AE2610 Introduction to Experimental Methods in Aerospace

Documents

Transcript of AE2610 Introduction to Experimental Methods in Aerospace