Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
1
Stanford UniversitySchool of Engineering
ENGINEERING 1N
THE NATURE OF ENGINEERING
READING
• Taylor, Chapters 4 and 5
2
Bike RampsBike Ramp Angles
Upper Ramp Lower RampTechnique Slope (°) Slope (°)tan 3.70 2.34tan* 2.12 3.23tan** 4.39 3.29tan* 4.57 3.26tan 3.576 3.327tan, sin, cos** 4.56 3.06tan** 4.60 3.22
Count 7 7Max 4.60 3.33Min 2.1 2.3
Mean 3.93 3.10Std Dev 0.91 0.35
C V 0.231 0.112
Slope (rad) Slope (rad)Mean 0.069 0.054
x
yz
q
q = tan-1 y
xÊ Ë
ˆ ¯
∂q∂x
=-y
x2 + y2 ∂q∂y
=x
x2 + y2
Error Propagation
3
Error Propagation
x
yz
q
dq =
-yx2 + y2
È
Î Í ˘
˚ ˙
2
dx 2 +x
x2 + y2È
Î Í ˘
˚ ˙
2
dy 2
dq £
-yx2 + y2 dx +
xx2 + y2 dy Bound
Independent Errors
RELATIVE ERROR IN TAN-1(y/x)
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50Tan-1(y/x) [rad]
dx/x = 0.01
dx/x = 0.1
4
RELATIVE ERROR IN q
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80q (rad)
—— dx/x = 0.1 dx/x = 0.01
cos-1(x/z)
sin-1(y/z)
tan-1(y/x)
RELATIVE ERROR IN q
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80q (rad)
—— dx/x = 0.1 dx/x = 0.01
cos-1(x/z)
sin-1(y/z)
tan-1(y/x)
5
Bike RampsBike Ramp Angles
Upper Ramp Lower RampTechnique Slope (°) Slope (°)tan 3.70 2.34tan* 2.12 3.23tan** 4.39 3.29tan* 4.57 3.26tan 3.576 3.327tan, sin, cos** 4.56 3.06tan** 4.60 3.22
Count 7 7Max 4.60 3.33Min 2.1 2.3
Mean 3.93 3.10Std Dev 0.91 0.35
C V 0.231 0.112
Slope (rad) Slope (rad)Mean 0.069 0.054
Measurement Uncertainty
• Quantificationw Bounds
Maximum/minimum values, instrument scale divisions
w Statistical measuresStandard deviation, s or s
6
Histograms
• Definedw Graphical presentation of a data set sorted by
magnitude, showing the relative proportion ofobservations of different magnitudes
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
True Value = 1.700
Descriptive Statistics
• Central Tendencyw Mean (Average)w Median Midpointw Mode Most Frequent
• Spreadw Standard Deviationw Variancew Coefficient of Variation
• Asymmetryw (Coefficient of) Skew
x = 1
nxi
i =1
n
Â
s =
1n -1
xi - x [ ]2
i =1
n
Â
CV =
sx
g =
n2
n - 1[ ] n - 2[ ]
xi - x [ ]3
i= 1
n
Âs3
†
Var = s2 Most commonuncertainty
index(as a multiple)
7
Raw Data Set 1
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
Data Set 1 Histogram
0
2
4
6
8
10
12
Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
8
Data Set 1 Histogram
0
2
4
6
8
10
12
14
16
Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 1 Histogram
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
9
Data Set 1 Histogram
0
10
20
30
40
50
60
1.35 1.55 1.75 1.95 2.15Bin Upper Limit (sec)
Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Histograms
• Design Issuesw Bin size
10
Raw Data Set 2
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80Observation
Data Set 2 Histogram
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
11
Data Sets 1 & 2 Histograms
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Raw Data Sets 1 & 2
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
12
Data Sets 1 & 2 Normalized Histograms
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Histograms
• Design Issuesw Bin sizew Normalization
13
Data Sets 1 & 2 Cumulative Histograms
0
10
20
30
40
50
60
70
80
90
100
Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
Data Sets 1 & 2 Normalized Cumulative Histograms
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bin Upper Limit (sec)
Data Set 1 Data Set 2
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 2Max = 1.896Min = 1.432Median = 1.667Mean = 1.676Std Dev = 0.104CV = 0.062Skew = 0.067
14
Cumulative Histograms
• The information in a histogram can be rearranged to createa cumulative histogram.w Bar height = Fraction of observations ≤ bin limitw Bar height = Sum of all histogram bar heights for all bins ≤ bin limitw There is a reciprocal relationship between the histogram and the
cumulative histogram, analogous to the relationship between thederivative and the integral
Histogram ¤ Cumulative histogram
Raw Data Set 3
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
15
Raw Data Sets 1 & 3
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
Data Sets 1 & 3 Histograms
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Data Set 1 Data Set 3
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 3Max = 1.984Min = 1.410Median = 1.689Mean = 1.689Std Dev = 0.110CV = 0.065Skew = 0.277
16
Raw Data Set 4
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
Raw Data Sets 1 & 4
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 10 20 30 40 50 60 70 80 90 100Observation
17
Data Sets 1 & 4 Histograms
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Data Set 1 Data Set 4
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 4Max = 2.728Min = 0.793Median = 1.707Mean = 1.692Std Dev = 0.330CV = 0.195Skew = 0.034
Raw Data Set 5
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
18
Raw Data Sets 1 & 5
0.00
0.50
1.00
1.50
2.00
2.50
0 10 20 30 40 50 60 70 80 90 100Observation
Data Sets 1 & 5 Histograms
0
5
10
15
20
25
30
35
40
Bin Upper Limit (sec)
Data Set 1 Data Set 5
Data Set 1Max = 2.030Min = 1.404Median = 1.718Mean = 1.708Std Dev = 0.116CV = 0.068Skew = 0.033
Data Set 5Max = 1.843Min = 1.198Median = 1.502Mean = 1.497Std Dev = 0.110CV = 0.073Skew = 0.038
19
Raw Data Set 6
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 10 20 30 40 50 60 70 80 90 100Observation
Raw Data Sets 4 & 6
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 10 20 30 40 50 60 70 80 90 100Observation
20
Data Sets 4 & 6 Histograms
0
5
10
15
20
Bin Upper Limit (sec)
Data Set 4 Data Set 6
Data Set 4Max = 2.728Min = 0.793Median = 1.707Mean = 1.692Std Dev = 0.330CV = 0.195Skew = 0.034
Data Set 6Max = 3.051Min = 0.986Median = 1.681Mean = 1.697Std Dev = 0.334CV = 0.197Skew = 0.938
E1N - The Nature of Engineering
Terman Pond Bags of Salt Histogram (2001 & 2003)
0
1
2
3
4
5
90 100 110 120 130 140 150 160 32050 Lb. Bags of Salt [Bin upper limit]
Max = 320Min = 99
Average = 136.2 BagsSt Dev = 54.6 Bags
CV = 0.40
21
E1N - The Nature of Engineering
Terman Pond Bags of Salt Histogram (2001 & 2003)
0
1
2
3
4
5
90 100 110 120 130 140 150 16050 Lb. Bags of Salt [Bin upper limit]
Max = 160Min = 99
Average = 123.1 BagsSt Dev = 20.8 Bags
CV = 0.17