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Machine Design Under Uncertainty
Outline
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Uncertainty in mechanical components
Why consider uncertainty
Basics of uncertainty
Uncertainty analysis for machine design
Examples
Conclusions
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Uncertainty in Mechanical Components
• A simply-supported beam has a diameter of
1.25 in. The deflection at 𝑥 = 10 in should be
less than δ =0.00375 in. Can the requirement
be met?
• Everything is modeled perfectly.
• In reality, the forces and dimensions are all
random.
• So is the deflection.
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Where Does Uncertainty Come From?
• Manufacturing impression
– Dimensions of a component
– Material properties
• Environment
– Loading
– Temperature
– Different users
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Why Consider Uncertainty?
• We know the true solution.
• We know the effect of uncertainty.
• We can make more reliable decisions.
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How Do We Model Uncertainty?
• We use probability distributions to model
parameters with uncertainty.
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Probabilistic Design
pdf
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Baseline Design
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50
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Noise
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Probability Distribution
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• With more samples, we can draw
a histogram.
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40
30
20
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• If y-axis is frequency and the
number of samples is infinity, we
get a probability density function
(PDF) 𝑓(𝑥).
• The probability of 𝑎 ≤ 𝑋 ≤ 𝑏.
Pr 𝑎 ≤ 𝑋 ≤ 𝑏 =
𝑏
𝑓
𝑎
0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
2
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1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
𝑥 𝑑𝑥
0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
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Normal Distribution
• 𝑋~𝑁(𝜇, 𝜎 2 )
• 𝐹 𝑥 = Pr 𝑋 < 𝑥 :
cumulative distribution
function (CDF)
• Pr 𝑎 < 𝑋 < 𝑏 = 𝐹 𝑏 −
𝐹 𝑎
• Pr 𝑋 < 𝑥 = 
𝑥−𝜇𝑌
𝜎𝑌
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
• Pr 𝑋 > 𝑥 = 1 − Pr 𝑋 <
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More about Standard Deviation 𝜎 (std)
• It indicates how data spread around the mean.
• It is always non-negative.
• High std means
– High dispersion
– High uncertainty
– High risk
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More Than One Random Variables
• If
–
–
–
–
𝑋𝑖 ~𝑁 𝜇𝑖 , 𝜎𝑖2
𝑋𝑖 𝑖 = 1,2, ⋯ , 𝑛 are independent
𝑌 = 𝑐0 + 𝑐1 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑛 𝑋𝑛
𝑐𝑖 𝑖 = 1,2, ⋯ , 𝑛 are constants.
• Then
– 𝑌~𝑁 𝜇𝑌 , 𝜎𝑌2
– 𝜇𝑌 = 𝑐0 + 𝑐1 𝜇1 + 𝑐2 𝜇2 + ⋯ + 𝑐𝑛 𝜇𝑛
– 𝜎𝑌 =
𝑐12 𝜎12 + 𝑐22 𝜎22 + ⋯ + 𝑐𝑛2 𝜎𝑛2
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Reliability
• Reliability is the ability of a component to
perform its intended function without failure.
• Reliability is measure by the probability of such
ability.
• 𝑅 = Pr 𝑔 𝐗 > 0}
– 𝐗: random variables
– 𝑔 ∙ : limit-state function
– If 𝑔 𝐗 <0, a failure occurs
• Probability of failure 𝑝𝑓 = Pr 𝑔 𝐗 < 0 = 1 − 𝑅
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First Order Second Moment Method (FOSM)
• Assume 𝑋𝑖 ~𝑁 𝜇𝑖 , 𝜎𝑖2 and are independent
• First order Taylor expansion
Y=𝑔 𝐗 ≈ 𝑔 𝛍 +
𝑛 𝜕𝑔
𝑖=1 𝜕𝑋
𝛍 = (𝜇1 , 𝜇2 , ⋯ , 𝜇𝑛 )
• 𝑌~𝑁
𝜇𝑌 , 𝜎𝑌2
, 𝜇𝑌 = 𝑔 𝛍 ,
• 𝑝𝑓 = Pr 𝑔 𝐗 < 0 = 
𝑖
(𝑋𝑖 − 𝜇𝑖 )
𝜎𝑌2
=
𝜕𝑔 2 2
𝑛
𝜎𝑖
𝑖=1 𝜕𝑋
𝑖
−𝜇𝑌
𝜎𝑌
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Monte Carlo Simulation (MCS)*
A sampling-based simulation method
Distributions of
input variables
Step 1: Sampling of random variables
Generating samples of random variables
Samples of input
variables
Step 2: Numerical Experimentation
Evaluating performance function
Samples of output
variables
Analysis Model
X
Analysis Model
Y
Y=g(X)
Step 3: Statistic Analysis on model output
Extracting probabilistic information
Probabilistic characteristics
of output variables
*This topic is optional.
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Step 1: Sampling on random variables
• Generate samples of input random variables
according to their distributions.
• For example, for 𝑋𝑖 ~𝑁 𝜇𝑖 , 𝜎𝑖2 , samples can
be generated by Matlab.
– Matlab normrnd(𝜇𝑖 , 𝜎𝑖 , 1, 𝑁) produces a row
vector of 𝑁 random samples.
– Excel can also be used.
Step 2: Obtain Samples of Output
• Suppose N sets of random variables have been
generated
𝐱𝑖 = (𝑥𝑖1 , 𝑥𝑖2 , ⋯ , 𝑥𝑖𝑛 ), 𝑖 = 1,2, ⋯ , 𝑁
𝑁 is the number of simulations
• Then samples of output s are calculated a
𝑦𝑖 = 𝑔 𝐱𝑖
Step 3: Statistic Analysis on output
• Mean 𝜇𝑌 =
1
𝑁
𝑁
𝑖=1 𝑦𝑖
• Standard deviation 𝜎𝑌 =
1
𝑁−1
𝑁
𝑖=1
𝑁𝑓
• The probability of failure 𝑝𝑓 =
𝑁
𝑁𝑓 is the number of failures.
𝑁𝑓 = number of 𝑦𝑖 <0, 𝑖 = 1,2, ⋯ , 𝑁
𝑦𝑖 − 𝜇𝑖
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FORM vs MCS
• FORM is more efficient
• FORM may not be accurate when a limit-state
function is highly nonlinear
• MCS is very accurate if the sample size is
sufficiently large
• MCS is not efficient
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Example - FOSM
L
Py
Px
t
w
• 𝐿 = 100 in, 𝑡 = 2 in, 𝑤 = 4 in, 𝐸 = 30106 psi
• 𝐗 = (𝑃𝑥 , 𝑃𝑦 ), 𝑃𝑥 ~𝑁 500, 602 lb, 𝑃𝑦 ~𝑁 1000, 1002
lb, 𝑃𝑥 and 𝑃𝑦 are independent
• Allowable deflection 𝐷0 = 3 in
• 𝑌 = 𝑔 𝐗 = 𝐷0 −
4𝐿3
𝐸𝑤𝑡
𝑃𝑥 2
𝑤2
+
𝑃𝑦 2
𝑡2
• 𝑝𝑓 = Pr 𝑌 = 𝑔 𝐗 < 0}
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Example - FOSM
• 𝜇𝑌 = 𝑔 𝛍 =
4 1003
30106 4 2
•
𝜕𝑔
𝜕𝑃𝑥
•
𝜕𝑔
𝜕𝑃𝑦
𝑃𝑦 2
4𝐿3
𝑃𝑥 2
𝐷0 −
+ 2 =
2
𝐸𝑤𝑡
𝑤
𝑡
500 2
1000 2
+
= 0.6708
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=
4𝐿3 𝑃𝑥 1
−
𝐸𝑤𝑡 𝑤 4 𝐴
= −3.7268𝑒 − 03
=
4𝐿3 𝑃𝑦 1
−
𝐸𝑤𝑡 𝑡 4 𝐴
= −4.6585𝑒 − 04
where 𝐴 =
𝑃𝑥 2
𝑤2
+
3−
𝑃𝑦 2
𝑡2
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Example - FOSM
• 𝜎𝑌 =
𝜕𝑔 2 2
𝜎𝑃𝑥
𝜕𝑃𝑥
• 𝑝𝑓 = 
−𝜇𝑌
𝜎𝑌
+
=
𝜕𝑔
𝜕𝑃𝑦
2
𝜎𝑃2𝑦 = 0.2284
−0.6708
0.2284
−3
=
 −2.9367 = 1.70 × 10
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Example - MCS
L
Py
Px
t
w
• 𝐿 = 100 in, 𝑡 = 2 in, 𝑤 = 4 in, 𝐸 = 30106 psi
• 𝐗 = (𝑃𝑥 , 𝑃𝑦 ), 𝑃𝑥 ~𝑁 500, 1002 lb,
𝑃𝑦 ~𝑁 1000, 1002 lb, 𝑃𝑥 and 𝑃𝑦 are independent
• Allowable deflection 𝐷0 = 3 in
• 𝑔 𝐗 =
4𝐿3
𝐷0 −
𝐸𝑤𝑡
𝑃𝑥 2
𝑤2
+
𝑃𝑦 2
𝑡2
• 𝑝𝑓 = Pr 𝑔 𝐗 < 0}
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100 and 1000 Simulations
3000
3000
Number of failures = 0
2500
Number of failures = 1
2500
Number of simulations = 100
Number of simulations = 1000
Probability of failure = 0
Probability of failure = 0.001
1500
2000
Safe region
Py
Py
2000
Failure region
1500
1000
1000
500
500
0
100
200
300
400
500
Px
600
700
800
900
0
100
Safe region
200
300
Failure region
400
500
Px
600
700
800
900
1e5 Simulations
• More simulations, More accurate result
3000
0.08
Number of failures = 191
2500
Number of simulations = 100000
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Probability of failure = 0.00191
2000
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pdf
Py
0.05
1500
Failure region
Safe region
0.04
0.03
1000
0.02
500
0.01
0
-0.5
0
0.5
1
g
1.5
2
0
100
200
300
400
500
Px
600
700
800
900
Reliability –Based Design (RBD)
Design without
considering uncertainty:
Low reliability
Nominal design point
x2
Failure Region
Safe Region
x1
Actual design points
Design with considering
uncertainty: high
reliability
x2
Failure Region
Safe Region
x1
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RBD
• RBD ensures that a design has the probability
of failure less than an acceptable level, and
• therefore ensures that events that lead to
catastrophe are extremely unlikely.
• RBD is achieved by maximizing cost and
maintaining reliability at a required level.
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Conclusions
• For important mechanical components in
important applications,
• a factor of safety may not be sufficient to
account for uncertainties;
• it is imperative to consider reliability.
• Uncertainty can be modeled probabilistically.
• Reliability can be estimated by FOSM and
MCS.
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