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Design with uncertainty. Prof. Dr. Vasilios Spitas. What is uncertainty?. The deviation (u) of an anticipated result ( μ ) within a margin of confidence (p). How familiar are we with uncertainty?. Hesitation Chance Luck Ambiguity Expectation. Error Probability Risk Reliability - PowerPoint PPT Presentation
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AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Design with uncertainty
Prof. Dr. Vasilios Spitas
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
What is uncertainty?
• The deviation (u) of an anticipated result (μ) within a margin of confidence (p)
:p μ u x μ u
1 :p x μ u x μ u
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
How familiar are we with uncertainty?
• Hesitation• Chance• Luck• Ambiguity• Expectation
• Error• Probability• Risk• Reliability• ToleranceQUALIT
ATIVE
QUANTITATIVE
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Quantitative assessment requires …
• Knowledge of the real problem• BOUNDARY CONDITIONS• Knowledge of the physical laws / interactions• CONSTITUTIVE EQUATIONS & CONSTANTS• Solvable / treatable formulation• MODEL• Solution• MATHEMATICS
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Discrete Continuous
Mean value
Standard deviation
• Discrete and continuous probability distribution functions
• Metrics:
Basic mathematical background
1
1 n
ii
x xn
X
μ xp x dx
2
1
1 n
ii
s x xn
2
X
σ x μ p x dx
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Normal distribution 2
2
1( ) exp22x μ
f xσσ π
Basic mathematical background
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Weibull distribution
Basic mathematical background
1
( ) expk kk x xf x
λ λ λ
1Γ 1μ λk
1
ln2 kσ λ
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• The sample / measurement set
• Follows the statistical distribution
• If and only if the likelihood function
• Satisfies the equation
From data sets to distribution functions
1 2, , , nX x x x
1 2, , , mf x α a a
1 1 11
, , , , ,n
m n m ii
L a a x x f a a x
ln 0L
Maximum Likelihood
Method
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• State a null hypothesis
• And an alternative hypothesis
• Such that either Ho or H1 are true. Then verify the null hypothesis using
Z – tests
Student’s tests
F – tests (ANOVA)
Chi – square tests
Statistical hypothesis testing
1 2, , ,o nH x x x
1 1 2, , , nH x x x
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• A random sample of size n
• Coming from a population of unknown distribution function with mean value (μ) and standard deviation (σ), has an average which follows the normal distribution with mean value:
• And standard deviation:
Central limit theorem
1 2, , , nx x x
averageμ μ
averageσσn
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Linking uncertainty with standard deviation
Less strict … … More strict
Confidence level
68.3% 95.4% 99.7% 99.99966%
Uncertainty 1σ 2σ 3σ 6σ
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• The uncertainty of a function
• With arguments xi and uncertainty ui each, is calculated as:
Combined uncertainty
1 2, , , nf x x x
2
1
n
fii i
fu ux
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Dimensional toleranceThe acceptable uncertainty of a dimension
Tolerancing in Embodiment Design
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Geometrical toleranceThe acceptable uncertainty of a feature form - location
Tolerancing in Embodiment Design
Form
Form
Form
Form
Form
Orientation
Orientation
Orientation
Orientation
Position
Position Position
Runout
Runout
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Understanding tolerancing
Tolerancing in Embodiment Design
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Communicating a function through tolerancing
Tolerancing in Embodiment Design
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Communicating functions through tolerancing
Tolerancing in Embodiment Design
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• A 50mm long 50 piezostack is formed by assembling 50 identical PZT disks, each 1mm in thickness and with a parallelism tolerance of 0.02mm. What is the resulting parallelism of the assembled stacks?
Example of combined tolerance calculation
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Let Δti be the deviation in parallelism of part i (i=1-50)
• The piezostack length is the sum of the individual thicknesses of the parts ti
• The requested uncertainty would then be:250
1
50 Δ 7.07 0.02 0.14fi ii i
fu u t mmx
Example of combined tolerance calculation
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
If we are sure that none of the parts exceeds the tolerance …
… then where is the uncertainty ?
Tolerance zone
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
• Analysisbreak the complex part into two or more simpler parts
• Synthesiscombine two or more parts into one monolithic part
• Inversionfemale geometries to male geometriescompression to tensioninternal features to external features
• Constraint control
Methods for reducing uncertainty in engineering design
AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands
Thank you for your attention
Good luck with the workshop assignments