Transcript Introduction

Uncertainty in Engineering Introduction
Jake Blanchard
Fall 2010
Uncertainty Analysis for Engineers
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Instructor
Jake Blanchard
 Engineering Physics
 143 Engineering Research Building
 [email protected]
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Course Web Site

eCOW2
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Uncertainty Analysis for Engineers
Course Goals:
 Students completing this course should be able to:
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◦ 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
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Grading
Homework – 30%
 1 Midterm – 30%
 Final Project – 40%

◦ Due Thursday, December 21, 2010
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Office Hours
Come see me any time
 Email or call if you want to make sure I’m
available
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Topics
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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
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Statistical Inferences
Determining Distribution Models
◦ Goodness of Fit
◦ Software Solutions
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Regression and Correlation
Sensitivity Analysis
Bayesian Approaches
Engineering Applications
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References
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Uncertainty: 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 – Vose
Probabilistic Techniques in Exposure Assessment
– Cullen & Frey (on reserve)
Statistical Models in Engineering – Hahn &
Shapiro (on reserve)
Probability Concepts in Engineering – Ang & Tang
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Uncertainty in Engineering
Engineers 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
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Types of Uncertainty
Aleatory – uncertainty arising due to
natural variation in a system
 Epistemic – uncertainty due to lack of
knowledge about the behavior of a
system
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An Example
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Aleatory – 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.
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Epistemic – weather
◦ We’re never quite sure what tomorrow’s weather
will be like, but our ability to predict has
improved
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Some Examples
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Some Examples
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Some Examples
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Some Examples
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How Do We Deal With This?
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Consider design of a diving board:
3
PL

3 EI
PLt

2I
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Diving Board
We need to get stiffness right to achieve
desired performance
 We need to make sure board doesn’t fail
 Options:
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◦ 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)
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Sensitivity vs. Uncertainty
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Consider the system pictured below:
Fsin(t)
k
k
m
k
m
x1

F1  1
1
 2

x1 
 2
2
2 
2m  1   31   
k
 
m
2
1
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Sensitivity
Suppose 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 2.45
 Since the driving frequency is 1, we should
be safe
 To check, computing x 1 gives 0.6*F1
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Amplitude vs. Driving Freq. (F1=1)
20
15
10
5
0
-5
-10
-15
-20
-25
-30
0.5
1
1.5
2
2.5
3
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But What If Model Has Errors?
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There are errors in the model:
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Inputs might be wrong
Loads might be wrong
Driving frequency might be wrong
Etc.
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How Sensitive is the Result to
Variations in Inputs?
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Relative change in amplitude as a function
of relative change in 3 inputs (k=2; m=1)
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spring stiffness
mass
frequency
20
15
10
5
0
-5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
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0.4
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Sensitivity for Different Defaults
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k=10; m=1
1
spring stiffness
mass
frequency
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
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0.4
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Defaults Closer to Resonance
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k=1.1; m=1
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spring stiffness
mass
frequency
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4
2
0
-2
-4
-6
-8
-10
-12
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
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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?
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Uncertainty Analysis
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Assume all inputs have normal
distribution with standard deviation of 1%
of the mean
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Plot is
histogram
of
amplitudes
2
x 10
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.52
0.54
0.56
0.58
0.6
0.62
0.64
0.66
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0.68
0.7
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Uncertainty Analysis
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What if inputs have standard deviation of
5% of the mean
4
7
x 10
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
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10 Commandments of Analysis
1.
2.
3.
4.
5.
Define the problem clearly
Let problem drive analysis (not available
tools, for example)
Make the analysis as simple as possible
Identify all significant assumptions
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.)
Iteratively refine problem statement and
analysis
9. Document clearly and completely
10. Seek peer review
8.