Deal with Uncertainty in Dynamics

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Outline

• Uncertainty in dynamics• Why consider uncertainty• Basics of uncertainty• Uncertainty analysis in dynamics• Examples• Applications

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Uncertainty in Dynamics

• Example: given , find and • We found and • Everything is modeled perfectly.• In reality, , , and are all random.• So are the solutions.

– will fluctuate around .

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Where Does Uncertainty Come From?

• Manufacturing impression– Dimensions of a mechanism– 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.

– If we know uncertainty in the traffic to the airport, we can better plan our trip and have lower chance of missing flights.

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• Measure ten times, we get• rad/s

• How do we use the samples?• Average rad/s (use Excel)

How Do We Model Uncertainty?

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How Do We Measure the Dispersion?

• We could use and ), • But )=0. • To avoid 0, we use ; to have the same unit as

, we use • We actually use

Standard deviation: .• Now we have 0.16 rad/s for rad/s. (Use Excel)

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More about Standard Deviation (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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Probability Distribution• With more samples, we can draw a

histogram.

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

10

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• If y-axis is frequency and the number of samples is infinity, we get a probability density function (PDF) .

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

0.2

0.4

0.6

0.8

1

1.2

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• The probability of .

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Normal Distribution

• is called cumulative distribution function (CDF)

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

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1.2

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How to Calculate ?

• Use Excel– NORMDIST(x,mean,std,cumulative)– cumulative=true

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Example 1• Average speeds (mph) of 20

drivers from S&T to STL airport were recoded.

• The average speed is normally distributed.– Mean =? Std = ?– What is the probability that a person

gets to the airport in 2 hrs if d = 120 mile?

1 62.22 58.23 70.94 64.95 68.36 67.97 67.18 63.09 63.2

10 65.511 62.312 62.813 72.914 67.215 62.116 66.817 62.518 61.619 66.020 70.8

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Solution: Use Excel

• average speed– mph by AVERAGE– = 3.72 mph by STDEV

• For hr, mph0.92

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More Than One Random Variables

• are independent

• are constants.• Then

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Example 2• The spool has a mass of m = 100

kg and a radius of gyration. It is subjected to force P= 1000 N. If the coefficient of static friction at A is determine if the spool slips at point A.

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Assume no slipping: = m=ma (2) = m=0 (3) = r =m(4) where =0.25 m, r =0.4 m

a 

NFf

x

P

G

G

FBD KD

y

mg

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Solving (1)—(4): = =1000 = 40.0N ( where =0.04; we’ll use it later) N=981.0 N =0.15(981.0)=147.15 N( where =981; we’ll use it later) =40.0 N < maxF =147.15N The assumption is OK, and the spool will not slip.

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When Consider Uncertainty

• If N()=N(0.15, )PN()=N(1000, ) N • What is the probability that the spool will slip ?

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Let Y= If Y<0, the spool will slip. = == = =29.9581 N Pr{Slipping} = Pr{Y<0}=F(0, 107.15, 29.9581, True) =1.74 (Use Excel)

Engine Robust Design

minimize ( , )noise noisef

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Result Analysis• The mean noise reduced by 0.5%.• The standard deviation reduced by 62.5%.

50 55 60 65 70 750

0.02

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0.1

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Noise

pd

f

Probabilistic Design

Baseline Design

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Assignment

• On Blackboard.