Design with uncertainty

AMP-EMBODYERASMUS Intensive ProgrammeDelft 8-24 July 2013, Netherlands

Design with uncertainty

Prof. Dr. Vasilios Spitas


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


How familiar are we with uncertainty?

• Hesitation• Chance• Luck• Ambiguity• Expectation

• Error• Probability• Risk• Reliability• ToleranceQUALIT

ATIVE

QUANTITATIVE


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


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


Normal distribution 2

2

1( ) exp22x μ

f xσσ π



Weibull distribution


1

( ) expk kk x xf x

λ λ λ

1Γ 1μ λk

1

ln2 kσ λ


• 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


• 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


• 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


Linking uncertainty with standard deviation

Less strict … … More strict

Confidence level

68.3% 95.4% 99.7% 99.99966%

Uncertainty 1σ 2σ 3σ 6σ


• 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


• Dimensional toleranceThe acceptable uncertainty of a dimension

Tolerancing in Embodiment Design


• Geometrical toleranceThe acceptable uncertainty of a feature form - location


Form

Form

Form

Form

Form

Orientation

Orientation

Orientation

Orientation

Position

Position Position

Runout

Runout


• Understanding tolerancing



• Communicating a function through tolerancing



• Communicating functions through tolerancing



• 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


• 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


If we are sure that none of the parts exceeds the tolerance …

… then where is the uncertainty ?

Tolerance zone


• 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


Thank you for your attention

Good luck with the workshop assignments

Documents

Design with uncertainty