Transcript Lecture Notes

ME 322: Instrumentation
Lecture 8
February 3, 2016
Professor Miles Greiner
Lab 4, Propagation of Uncertainty, Maximum and likely, Power
Product, Examples
Announcements/Reminders
• All future HW: all plots must be generated and
clearly formatted by Excel, MatLab, or…
– No hand-drawn plots
• This week in lab
– Lab 3 Pressure Transmitter Calibration
– Please bring you Excel from L3PP, and the
instructions.
• For the rest of the week, groups will move to
pressure standards instead of moving standards to
pairs of groups.
– Should be faster for everyone.
• We will purchase more standards before next
year.
Strain Gages
𝑑𝑅𝑖
= 𝑆𝑖 𝜀𝑖 + 𝑆𝑇𝑖 ∆𝑇𝑖
𝑅𝑖
• Their electrical resistance changes by small amounts when
– They are strained (desired sensitivity) or
– Their temperature changes (undesired sensitivity)
• Solution:
– Subject “identical” gages to the same environment so
they experience the same temperature change and the
same temperature-associated resistance change.
– Incorporate them in a circuit that cancels-out the
temperature effect
Strain Gage Wheatstone Bridge
+
•
-
𝑉0
𝑉𝑠
=
𝑅1 𝑅3 −𝑅2 𝑅4
𝑅2 +𝑅3 𝑅1 +𝑅4
• If initially, R1I = R2I = R3I = R4I
R3
-
– Then VOI ~ 0
+
• Bridge output voltage VO will exhibit large changes compared to VOI
when Ri’s changes by small amounts
𝑉𝑂
𝑉𝑆
=
1
4
𝑑𝑅1
𝑅1
−
𝑑𝑅2
𝑅2
+
𝑑𝑅3
𝑅3
• Incorporate strain gages in the
•
𝑑𝑉0
𝑉𝑠
=
1
4
𝑑𝑅4
−
𝑅4
𝑑𝑅𝑖
bridge:
𝑅𝑖
= 𝑆𝜀𝑖 + 𝑆𝑇 ∆𝑇𝑖
𝑆 𝜀1 − 𝜀2 + 𝜀3 − 𝜀4 + 𝑆𝑇 ∆𝑇1 − ∆𝑇2 + ∆𝑇3 − ∆𝑇4
Beam in Bending: Half Bridge
ε3
+
ε2 = -ε3
•
𝑉0
𝑉𝑆
=
1
4
ε2 = -ε3
∆𝑇2 ≈ ∆𝑇3
𝑆 𝜀3 − 𝜀2 + 𝑆𝑇 ∆𝑇3 − ∆𝑇2
=
R3
+
1
𝑆(2𝜀3 )
4
𝑉0
1
= 𝑆𝜀3
𝑉𝑆
2
– Twice the output amplitude as quarter bring, with temperature
compensation
• Lab 4 Install and test stain gages on opposite sides of aluminum
and steel beams, and measure beam densities
• Lab 5 Measure Aluminum and Steel Elastic Moduli
Lab 4 Install Strain Gages on Aluminum
and Steel Beams (next week)
LT
W
T
L
• Install two gage on a Steel or Aluminum beams
– Top and bottom
– Video Instructions:
http://wolfweb.unr.edu/homepage/greiner/teaching/MECH322Instrumentation/L
abs/Lab%2004%20Install_Strain_Gage/Lab%2004%20Index.htm
• Measure gage electrical resistance
– Checks whether wire we connected correctly
– Un-deformed, under tension, compression, and when
cold
Lab 4 Install Strain Gages on Aluminum
and Steel Beams (next week)
LT
W
T
L
• Measure beam thickness T, width W,
– Partners use Vernier calipers and micrometer
– Best values: 𝑇 and 𝑊 (mean)
– Uncertainties: Use sample standard deviation: 𝑤𝑇 = 𝑠𝑇 , 𝑤𝑊 = 𝑠𝑊
• What are their confidence levels? __%
• Total length LT, distance between centers of gage and pin, L
– Use ruler, tape measure for 𝐿 𝑇 and 𝐿
– Uncertainties: half of smallest increment; 𝑤𝐿 =
• Confidence levels? __%
1
"
16
𝑜𝑟
1
",
32
𝑤𝐿𝑇 =
1
“
16
• Measure beam mass and its uncertainty: 𝑚 = 𝑚 ± 𝑤𝑚 𝑢𝑛𝑖𝑡𝑠 𝑃𝑚
Calculate Beam Density
• 𝜌=
𝑚
𝑉
=
𝑚
𝑊𝑇𝐿𝑇
• Will everyone in the class get the same density as:
– A textbook? Each other? If they measure many times?
• Why not?
– Different samples have different densities
– Experimental errors in measuring lengths and masses
(due to calibration errors and imprecision)
• How can we estimate the uncertainty in 𝜌 (𝑤𝜌 ) from
uncertainties in
– 𝐿 𝑇 (𝑤𝐿𝑇 ), 𝑇 (𝑤𝑇 ), 𝑊 (𝑤𝑊 ), and 𝑚 (𝑤𝑚 )
Propagation of Uncertainty
• A calculation based on uncertain inputs
– R = fn(x1, x2, x3, …, xn)
• For each input 𝑥𝑖 find (measure, calculate) the best
estimate for its value 𝑥𝑖 , its uncertainty 𝑤𝑥𝑖 = 𝑤𝑖
with its confidence-level (certainty, probability) pi
– 𝑥𝑖 = 𝑥𝑖 ± 𝑤𝑖 𝑝𝑖 𝑖 = 1,2, … 𝑛
• The best estimate for the results is:
– 𝑅 = 𝑓𝑛(𝑥1 , 𝑥2 , 𝑥3 ,…, 𝑥𝑛 )
• Find the confidence-interval for the result
– 𝑅 = 𝑅 ± 𝑤𝑅 (𝑝𝑅 )
• Find 𝑤𝑅 𝑎𝑛𝑑 𝑝𝑅
𝑥
Concept
𝑅
𝑅
𝑤𝑅𝑖
𝛿𝑅
𝛿𝑥𝑖
𝑤𝑖
𝑥𝑖
𝑥𝑖
𝑥𝑖
• Vary one input 𝑥𝑖 by a small amount (𝑤𝑖 ) while holding all the
others inputs constant (linear analysis)
• Uncertainty in 𝑅 due to a small change in 𝑥𝑖 (𝑤𝑖 ) alone is
–
𝑤𝑅𝑖 =
𝛿𝑅
𝛿𝑥𝑖 𝑥
𝑖
𝑤𝑖
(why use absolute value?)
All the 𝑤𝑖 = 𝑤𝑅𝑖 ’s contribute to wR
• How to combine them?
• Two ideas:
• If all the wi’s are the maximum possible uncertainty
in xi (100% confident the true value is within 𝑥𝑖 ±
𝑤𝑖 )
• Then the maximum possible uncertainty in 𝑅 is:
– 𝑤𝑅,𝑀𝐴𝑋 =
𝑛
𝑖=1
𝑤𝑅𝑖 =
𝑛
𝑖=1
𝛿𝑅
𝛿𝑥𝑖 𝑥
𝑖
𝑤𝑖
– Use absolute values because it’s not likely that all of the
derivatives will be positive
–
𝑅 = 𝑅 ± 𝑤𝑅,𝑀𝐴𝑋 (100%)
Likely Uncertainty
• Maximum possible error is overly pessimistic and
not useful since it’s not likely
– that all the actual measured values will have the
maximum possible deviation from the best estimate, or
– that they will all push the result in the same direction
• A “Likely” estimate (p < 100%) would be more
realistic and useful uncertainty estimate
– 𝑊𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 (pR < 100%) < 𝑊𝑅,𝑀𝐴𝑋 (pR = 100%)
Statistical Analysis Shows
(meaning: take my word for it)
• 𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 =
𝑛
𝑖=1
𝑤𝑅𝑖
2
=
𝑛
𝑖=1
𝛿𝑅
𝛿𝑥𝑖 𝑥
𝑖
𝑤𝑖
2
• In this expression
– Confidence level for all the 𝑤𝑖 ’s, pi (i = 1, 2,…, n) must be the
same
– Confidence level of 𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 , pR = pi is the same at the 𝑤𝑖 ’s
– All errors must be uncorrelated
• Not biased by the same calibration error
𝑥
Rectangle Example
𝑏
𝑎
𝑤𝑏
A = ab
𝑤𝑎
• Given
– 𝑎 = 𝑎 + 𝑤𝑎 = 4.12 ± 0.1 𝑐𝑚 = ___? __ ± 0.1 𝑐𝑚 (68%)
– 𝑏 = 𝑏 + 𝑤𝑏 = 12. 31 ± 0.1 𝑐𝑚 = ___? __ ± 0.1 𝑐𝑚(68%)
• Find 𝐴 = 𝐴 + 𝑤𝐴
• Find best estimate 𝐴
– How many digits are significant? (find 𝑤𝐴 )
• Which length’s uncertainty has a larger contribution to the area uncertainty?
Work on the white board
• Identify:
– Are we trying to find the Maximum or Likely
Uncertainty?
– Are all the certainty levels the same?
–n=?
A Simple method to propagate
uncertainties in Power Product calculations
• “Power Product”
– 𝑅 = 𝑎𝑋 𝑏 𝑌 𝑐 …
𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 are constants
• Confidence Intervals of Inputs
– 𝑋 = 𝑋 ± 𝑤𝑋
𝑌 = 𝑌 ± 𝑤𝑌 …
• Best Estimate
– 𝑅 = 𝑎𝑋 𝑏 𝑌 𝑐 …
• Likely Uncertainty
– 𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 =
–
2
𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 =
=
𝑛
𝑖=1
𝛿𝑅
𝛿𝑥𝑖 𝑥
𝑖
2
𝑤𝑖
2
𝜕𝑅
𝑤
𝜕𝑋 𝑋
2
𝑏−1
𝑐
𝑏𝑎𝑋
𝑌 𝑤𝑋
+
+
2
𝜕𝑅
𝑤
𝜕𝑌 𝑌
𝑐𝑎 𝑋 𝑏 𝑌 𝑐−1
𝑤𝑌
2
Fractional Uncertainty
• Divide Uncertainty 𝑤𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 2 by 𝑅2
–
–
𝑊𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 2
𝑅2
=
𝑊𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 2
𝑅
𝑏𝑎𝑋 𝑏−1 𝑌 𝑐 𝑤𝑋
2
𝑎𝑋 𝑏 𝑌 𝑐
=
𝑤𝑋 2
𝑏
𝑋
+
2
+
𝑐𝑎𝑋 𝑏 𝑌 𝑐−1 𝑤𝑌
𝑤𝑌 2
𝑐
𝑌
𝑎𝑋 𝑏 𝑌 𝑐
2
2
General Power Product Uncertainty
𝑛
𝑒𝑖
𝑥
𝑖=1 𝑖
• 𝑅=𝑎
where a and ei are constants
• The likely fractional uncertainty in the result is
–
𝑊𝑅,𝐿𝑖𝑘𝑒𝑙𝑦 2
𝑅
=
𝑛
𝑖=1
𝑊𝑖 2
𝑒𝑖
𝑥𝑖
– Square of fractional error in the result is the sum of the
squares of fractional errors in inputs, multiplied by their
exponent.
• The maximum fractional uncertainty in the result is
–
𝑊𝑅,𝑀𝑎𝑥
𝑅
=
𝑛
𝑖=1
𝑊𝑖
𝑒𝑖
𝑥𝑖
(100%)
– We don’t use maximum errors much in this class
Repeat Rectangle Example using
Power Product
Rule
𝑏
𝑤
𝑏
𝑎
A = ab
𝑤𝑎
• Given
– 𝑎 = 𝑎 + 𝑤𝑎 = 4.12 ± 0.1 𝑐𝑚 = 4.1 ± 0.1 𝑐𝑚
– 𝑏 = 𝑏 + 𝑤𝑏 = 12. 31 ± 0.1 𝑐𝑚 = 12.3 ± 0.1 𝑐𝑚
• Find 𝐴 = 𝐴 + 𝑤𝐴
• Work on board
Lab 4: Calculate Beam Density
LT
W
T
L
• 𝜌=
𝑚
𝑉
=
𝑚
𝑊𝑇𝐿𝑇
• Measure and estimate 95%-confidence-level uncertainties of
–
–
–
–
𝑚 = 𝑚 ± 𝑤𝑚 𝑔𝑚 95%
𝑊 = 𝑊 ± 𝑤𝑊 𝑖𝑛𝑐ℎ 95%
𝑇 = 𝑇 ± 𝑤𝑇 𝑖𝑛𝑐ℎ 95%
𝐿 𝑇 = 𝐿 𝑇 ± 𝑤𝐿𝑇 𝑖𝑛𝑐ℎ 95%
• Best estimate
– 𝜌=
𝑚
𝑊𝑇𝐿𝑇
• Power product? (yes or no)
–
𝑤𝜌 2
𝜌
= Fill in blank
– If all the 𝑝𝑖 = 0.95, then 𝑝𝜌 = ?
• How to find 𝑤𝑖 with 𝑝𝑖 = 0.95?