Farizal
Efstratios Nikolaidis
SAE 2007 World Congress
1

Calculation of Upper and Lower Reliabilities of System with
Dynamic Vibration Absorber
2

Challenges in Reliability Assessment of
Engineering Systems:
– Scarce data, poor understanding of physics
• Difficult to construct probabilistic models
• No consensus about representation of uncertainty in probabilistic models
– Calculations for reliability analysis are expensive
3

(continued)
• Modeling uncertainty in probabilistic models
Probability
• Second-Order Probability: Parametric family of probability distributions. Uncertain distribution parameters,

, are random variables with PDF f
Θ
( θ )
• Reliability - random variable
.
1
CDF
0.5
0
0.97
0.98
R(

)
0.99
E ( R )

1
  f Θ (
θ
)[ 
F
1 f
X
( x ,
θ
) d x] d
θ
4

(continued)
Interval Approach to Model Uncertainty
Given ranges of uncertain parameters find minimum and maximum reliability
.
R
R
1
CDF
0.5
0
0.97
0.98
0.99
1
– Finding maximum or minimum reliability: Nonlinear
Programming, Monte Carlo Simulation, Global Optimization
– Expensive – requires hundreds or thousands reliability analyses
5

• Develop efficient Monte-Carlo simulation approach to find upper and lower bounds of Probability of Failure (or of Reliability) given range of uncertain distribution parameters
6

General formulation of global optimization problem
Max (Min)

Such that:
PF (

)
θ 
[
θ
,
θ
]
7

• Monte-Carlo simulation
– Select a sampling PDF for the parameters θ and generate sample values of these parameters. Estimate the reliability for each value of the parameters in the sample. Then find the minimum and maximum values of the values of the reliabilities.
– Challenge: This process is too expensive
8

• Importance Sampling
PF

1 n i n


1
I ( x i
) g f
X
( x i
X
, θ )
( x i
, θ )
True PDF
Sampling PDF
9

• If we estimate the reliability for one value the uncertain parameters θ using Monte-Carlo simulation, then we can find the reliability for another value θ’ very efficiently.
• First, calculate the reliability, R ( θ ), for a set of parameter values, θ. Then calculate the reliability, R ( θ’ ), for another set of values θ’ as follows:
If R ( θ )

1

1 n
 i
I i
( x i
) f
X g s
X
( x
( x i i
,
,
θ
)
θ
)
( 1 ) then
R ( θ 
)

1

1 n
 i
I i
( x i
) f g
X s
X
( x i
( x i
,
,
θ 
)
θ
)
(2)
10

• Idea: When calculating R (
 ’), use the same values of the failure indicator function as those used when calculating
R (

).
• We only have to replace the PDF of the random variables, f
X
( x , θ ), in eq. (1) with f
X
( x , θ ’).
• The computational cost of calculating R (
 ’) is minimal because we do not have to compute the failure indicator function for each realization of the random variables.
11

Using Extreme Distributions to Estimate
Upper and Lower Reliabilities
PDF
PDF of smallest reliability in sample
Parent PDF
(Reliabilities in a sample follow this
PDF)
Reliability
If we generate a sample of N values of the uncertain parameters θ, and estimate the reliability for each value of the sample, then the maximum and the minimum values of the reliability follow extreme type III probability distribution.
12

Algorithm for Estimation of Lower and Upper
Probability Using Efficient Reliability Reanalysis
Information about
Uncertain Distribution
Parameters
Reliability
Analysis
Repeated
Reliability
Reanalyses
Path A
Estimate of Global Min and Max Failure
Probabilities
Path B
Fit Extreme Distributions
To Failure Probability
Values
Estimate of Global Min
And Max Failure Probability
From Extreme Distributions
13

Path B: Estimation of Lower and Upper Probabilities
14

Example: Calculation of Upper and Lower Failure
Probabilities of System with Dynamic Vibration
Absorber m ,
 n2
Dynamic absorber
Normalized system amplitude y
F = cos (
 e t )
M ,
 n1
Original system
15

• Evaluate the accuracy and efficiency of the proposed approach
• Determine the effect of the sampling distributions on the approach
• Assess the benefit of fitting an extreme probability distribution to the failure probabilities obtained from simulation
16

Displacement
β
2
β
1
17

• Calculation of failure probability is difficult
• Failure probability sensitive to mean values of normalized frequencies
• Failure probability does not change monotonically with mean values of normalized frequencies. Therefore, maximum and minimum values cannot be found by checking the upper and lower bounds of the normalized frequencies.
18

Max (Min)

R (

)
Such that : 0.9 ≤  i
≤ 1.1, i = 1, 2
0.05 ≤  i
≤ 0.2,
0 ≤ R (
 ) ≤ 1 i = 1, 2
 i
: mean values of normalized frequencies
 i
: standard deviations of normalized frequencies
19

PF max vs. number of replications per simulation ( groups of failure probabilities ( N ), and failure n ), probabilities per group ( m )
0.37
0.35
0.33
True value of PF max
2000 replications
5000 replications
10000 replications
0.31
0.29
0.27
0.25
36*25 120*25
N*m
36*1000
2000
5000
10000
Target PFmax
120*1000
20

36
120
N
Comparison of PF min and PF max
True PF max
=0.332
for n = 10,000
25
1000
25 m
1000
Proposed Method with ERR
PF min
(

PFmin
)
PF max
(

PFmax
)
0.03069
(0.0021)
0.02333
(0.0016)
0.03069
(0.0021)
0.02333
(0.0016)
0.27554
(0.0144)
0.30982
(0.0190)
0.3008
(0.0170)
0.32721
(0.0200)
PF min
(

PFmin
)
0.032
(0.0017)
0.0251
(0.0016)
0.032
(0.0018)
0.0239
(0.0016)
MC
PF max
(

PFmax
)
0.2763
(0.0045)
0.3106
(0.0046)
0.3004
(0.0046)
0.3249
(0.0047)
21

Effect of Sampling Distribution on PF max
PFmax for n = 10000
0.31
0.3
0.29
0.28
0.27
0.34
0.33
0.32
36*25
Monte
Two sampling distributions
One sampling distribution
Carlo
120*25
N*m
36*1000 single sampling bisampling
M C
True Value
120*1000
22

CPU time for simulation with n = 10000
N
36
120
CPU Time (sec) m
25
1000
25
1000
Proposed
Method with
ERR
2.70
100
8.61
342
MC
151
6061
503
20198
23

Fitted extreme CDF of maximum failure probability vs. data
N =120, m =1000, n =10000
Maximum Case: 120*1000*10K
1.2
1.0
Fitted, ERR
Fitted MC
0.8
0.6
0.4
0.2
M cmax
Dat a _M C
Ismax
Dat a_IS
0.0
0.
26
0.
26
4
0.
26
8
0.
27
2
0.
27
6
0.
28
0.
28
4
0.
28
8
0.
29
2
0.
29
6
PF
0.
3
0.
30
4
0.
30
8
0.
31
2
0.
31
6
24

• The proposed approach is accurate and yields comparable results with a standard Monte
Carlo simulation approach.
• At the same time the proposed approach is more efficient; it requires about one fiftieth of the CPU time of a standard Monte Carlo simulation approach.
• Sampling from two probability distributions improves accuracy.
• Extreme type III distribution did not fit minimum and maximum values of failure probability
25