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Uncertainty in Engineering - Introduction. Jake Blanchard Fall 2010. Instructor. Jake Blanchard Engineering Physics 143 Engineering Research Building [email protected]. Course Web Site. eCOW2. Uncertainty Analysis for Engineers. Course Goals: - PowerPoint PPT Presentation
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Uncertainty Analysis for Engineers 1
Uncertainty in Engineering - IntroductionJake BlanchardFall 2010
Uncertainty Analysis for Engineers 2
InstructorJake BlanchardEngineering Physics143 Engineering Research
Uncertainty Analysis for Engineers 3
Course Web SiteeCOW2
Uncertainty Analysis for Engineers 4
Uncertainty Analysis for Engineers
Course Goals: Students completing this course should be able
to:◦ create probability distribution functions for model
inputs◦ determine analytical solutions for output distribution
functions when the inputs are uncertain◦ determine numerical solutions for these same output
distribution functions◦ apply these techniques to practical engineering
problems◦ make engineering decisions based on these
uncertainty analyses
Uncertainty Analysis for Engineers 5
GradingHomework – 30%1 Midterm – 30%Final Project – 40%
◦Due Thursday, December 21, 2010
Uncertainty Analysis for Engineers 6
Office HoursCome see me any timeEmail or call if you want to make
sure I’m available
Uncertainty Analysis for Engineers 7
Topics Introduction to Engineering Uncertainty and Risk-Based
Decision Making Review of Probability and Statistics Probability Distribution Functions and Cumulative
Distribution Functions Multiple Random Variables (joint and conditional probability) Functions of Random Variables (analytical methods) Numerical Models
◦ Monte Carlo◦ Commercial Software
Statistical Inferences Determining Distribution Models
◦ Goodness of Fit◦ Software Solutions
Regression and Correlation Sensitivity Analysis Bayesian Approaches Engineering Applications
Uncertainty Analysis for Engineers 8
ReferencesUncertainty: A Guide to Dealing With
Uncertainty in Quantitative Risk and Policy Analysis - Morgan & Henrion
Probability, Statistics, and Decision for Civil Engineers – Benjamin & Cornell
Risk Analysis: A Quantitative Guide – VoseProbabilistic Techniques in Exposure
Assessment – Cullen & Frey (on reserve)Statistical Models in Engineering – Hahn &
Shapiro (on reserve)Probability Concepts in Engineering – Ang &
Tang
Uncertainty Analysis for Engineers 9
Uncertainty in EngineeringEngineers apply scientific and
mathematical principles to design, manufacture, and operate structures, machines, processes, systems, etc.
This entire process brings with it uncertainty and risk
We must understand this uncertainty if we are to properly account for it
Uncertainty Analysis for Engineers 10
Types of UncertaintyAleatory – uncertainty arising due
to natural variation in a systemEpistemic – uncertainty due to
lack of knowledge about the behavior of a system
Uncertainty Analysis for Engineers 11
An ExampleAleatory – radioactive decay
◦How long will it take for half of a sample to decay?
◦When will a particular atom decay?◦Decay has an intrinsic uncertainty. No
knowledge will help to reduce this uncertainty.
Epistemic – weather◦We’re never quite sure what tomorrow’s
weather will be like, but our ability to predict has improved
Uncertainty Analysis for Engineers 12
Some Examples
Uncertainty Analysis for Engineers 13
Some Examples
Uncertainty Analysis for Engineers 14
Some Examples
Uncertainty Analysis for Engineers 15
Some Examples
Uncertainty Analysis for Engineers 16
How Do We Deal With This?Consider design of a diving
board:
IPLtEIPL
2
3
3
Uncertainty Analysis for Engineers 17
Diving BoardWe need to get stiffness right to achieve
desired performanceWe need to make sure board doesn’t failOptions:
◦Use worst-case properties and loads and small safety factor
◦Use average properties and large safety factor
◦Spend more on quality control for materials and manufacturing (still have uncertainty in loads)
Uncertainty Analysis for Engineers 18
Sensitivity vs. UncertaintyConsider the system pictured
below:
x1
m mk k k
Fsin(t)
mkmFx
21
221
221
11 3
112
Uncertainty Analysis for Engineers 19
SensitivitySuppose we have a design (k=2,
m=1, =1) and we want to see how far we are from resonance
Resonant frequencies are 1 and 1.73 1
Or 1.41 and 2.45Since the driving frequency is 1,
we should be safeTo check, computing x 1 gives
0.6*F1
Uncertainty Analysis for Engineers 20
Amplitude vs. Driving Freq. (F1=1)
0.5 1 1.5 2 2.5 3
-30
-25
-20
-15
-10
-5
0
5
10
15
20
Uncertainty Analysis for Engineers 21
But What If Model Has Errors?There are errors in the model:
◦Inputs might be wrong◦Loads might be wrong◦Driving frequency might be wrong◦Etc.
Uncertainty Analysis for Engineers 22
How Sensitive is the Result to Variations in Inputs?Relative change in amplitude as
a function of relative change in 3 inputs (k=2; m=1)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-5
0
5
10
15
20
25
spring stiffnessmassfrequency
Uncertainty Analysis for Engineers 23
Sensitivity for Different Defaultsk=10; m=1
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
spring stiffnessmassfrequency
Uncertainty Analysis for Engineers 24
Defaults Closer to Resonancek=1.1; m=1
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-12
-10
-8
-6
-4
-2
0
2
4
6
8
spring stiffnessmassfrequency
Uncertainty Analysis for Engineers 25
How Much Variation Do We Expect?The final question is, how much
variation do we expect in these inputs?
Can we control variation in spring stiffness and mass?
What about controlling the frequency?
Uncertainty Analysis for Engineers 26
Uncertainty AnalysisAssume all inputs have normal
distribution with standard deviation of 1% of the mean
0.52 0.54 0.56 0.58 0.6 0.62 0.64 0.66 0.68 0.70
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Plot is histogram of amplitudes
Uncertainty Analysis for Engineers 27
Uncertainty AnalysisWhat if inputs have standard
deviation of 5% of the mean
0 0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7x 10
4
10 Commandments of Analysis1. Define the problem clearly2. Let problem drive analysis (not
available tools, for example)3. Make the analysis as simple as
possible4. Identify all significant
assumptions5. Be explicit about decision
criteria
10 Commandments (cont.)
6. Be explicit about uncertainties◦Technical, economic, and political quantities◦Functional form of models◦Disagreement among experts
7. Perform sensitivity and uncertainty analysis◦Which uncertainties are important◦Sensitivity=what is change in output for given
change in input◦Uncertainty=what is best estimate of output
uncertainty given quantified uncertainty in inputs
10 Commandments (cont.)8. Iteratively refine problem
statement and analysis9. Document clearly and
completely10.Seek peer review