deal with uncertainty in dynamics
DESCRIPTION
Deal with Uncertainty in Dynamics. Outline. Uncertainty in dynamics Why consider uncertainty Basics of uncertainty Uncertainty analysis in dynamics Examples Applications. Uncertainty in Dynamics. Example: given , find and We found and Everything is modeled perfectly. - PowerPoint PPT PresentationTRANSCRIPT
Deal with Uncertainty in Dynamics
2
Outline
• Uncertainty in dynamics• Why consider uncertainty• Basics of uncertainty• Uncertainty analysis in dynamics• Examples• Applications
3
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 .
4
Where Does Uncertainty Come From?
• Manufacturing impression– Dimensions of a mechanism– Material properties
• Environment– Loading– Temperature– Different users
5
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.
6
• Measure ten times, we get• rad/s
• How do we use the samples?• Average rad/s (use Excel)
How Do We Model Uncertainty?
7
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)
8
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
9
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
20
30
40
50
60
• 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
1.4
1.6
1.8
2
• The probability of .
10
Normal Distribution
• is called cumulative distribution function (CDF)
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
1.4
1.6
1.8
2
11
How to Calculate ?
• Use Excel– NORMDIST(x,mean,std,cumulative)– cumulative=true
12
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
13
Solution: Use Excel
• average speed– mph by AVERAGE– = 3.72 mph by STDEV
• For hr, mph0.92
14
More Than One Random Variables
• are independent
• are constants.• Then
15
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.
16
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
17
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.
18
When Consider Uncertainty
• If N()=N(0.15, )PN()=N(1000, ) N • What is the probability that the spool will slip ?
19
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
20
Result Analysis• The mean noise reduced by 0.5%.• The standard deviation reduced by 62.5%.
50 55 60 65 70 750
0.02
0.04
0.06
0.08
0.1
0.12
Noise
pd
f
Probabilistic Design
Baseline Design
21
22
Assignment
• On Blackboard.