ae2610 introduction to experimental methods in aerospace
TRANSCRIPT
AE2610Introduction to
Experimental Methods in Aerospace
Measurement Systems & Plotting
C.V. Di Leo
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Sensing/Measurement Systems
2
• Measurements• Absolute• Relative/calibrated
• Sensing/recording• Human based• Mechanical• Electrical
(Digital) Data Acquisition Systems
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• 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)
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Resolution = 50 psiDyn. Range= 5000/50
= 100
Sensitivity= 5000 psi/270°= 18.5 psi/degree
0° 270°
Computer Data Acquisition
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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)
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Graphing
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Graphing• Graphs are one of the most effective ways to
communicate experimental results• However, graphs are often constructed poorly.
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• 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
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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...
Good:• y=values on log scale
• x=every 20 years
• Circle symbols
• Linear trend...
Poor:• y=log()
• x=too many values
• No data symbols
• What does line mean?
Poor:• y=log()
• x=too many values
• No data symbols
• What does line mean?
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...
Good:• y=values on log scale
• x=every 20 years
• Circle symbols
• Linear trend...
Poor:• y=log()
• x=too many values
• No data symbols
• What does line mean?
Poor:• y=log()
• x=too many values
• No data symbols
• What does line mean?
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
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Linear• Both axes are linear• Symbols for data and lines
for trend/fits
Basic Types of Cartesian Plots
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• 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.
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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
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Spring 2001 AE 3145 Measurements Lab 4
Polar Chart
• Useful for showing angular effects
Three-Dimensional Plots• Show variations along three dimensions.
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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
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Spring 2001 AE 3145 Measurements Lab 5
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.
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Spring 2001 AE 3145 Measurements Lab 6
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• y=1 + 2.5/x• Asymptotic behavior• Hyperbolic behavior
Nonlinear Axes• y=same; x →1/x• Slope = 2.5• Intercept = 1.0
Nonlinear Axes• y=same; x →1/x• Slope = 2.5• Intercept = 1.0
Spring 2001 AE 3145 Measurements Lab 6
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• y=1 + 2.5/x• Asymptotic behavior• Hyperbolic behavior
Nonlinear Axes• y=same; x →1/x• Slope = 2.5• Intercept = 1.0
Nonlinear Axes• y=same; x →1/x• Slope = 2.5• Intercept = 1.0
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.
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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.
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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.
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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
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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?
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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
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Linear regression continued
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
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Linear regression continued
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
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