Reliability Modelling

Series Systems

Parallel Systems

Bayesian Testing

Design Verification

Design for Six Sigma roadmap

HoQ1￫Boundary￫HoQ2￫P-diagram￫DFMEA￫PFMEA￫SCIF￫Control plan

HoQ1￫Boundary￫HoQ2￫P-diagram￫DFMEA￫PFMEA￫SCIF￫Control plan

HoQ1￫Boundary￫HoQ2￫P-diagram￫DFMEA￫PFMEA￫SCIF￫Control plan

HoQ1￫Boundary￫HoQ2￫P-diagram￫DFMEA￫PFMEA￫SCIF￫Control plan

System validation plan

System verification plan

Sub-system verification plan

Component verification plan

Time line

System

Sub-system

Component

System

Sub-system

Slide 6

What do I Verify & Validate?

Slide 7

Pool is 50 metres long

500 metres

DfSS Process

Boundarydiagram

Parameter diagram

DesignFMEA

SCIF

HoQ #1

ProcessFMEA

SCIFManufacturingcontrol plan

Field performance

Project goals

System boundary diagram

Slide 9

Anglemechanism

Velocitymechanism

Support

Barrel

Distance(475 – 525 m)

Angle43o – 47o

Velocity (m/s)75.5 – 79.2

Gravitationalacceleration(m/sec2)9.81 – 9.82

Sound(100 – 120 dB)

Height(> 90 m)

Soundmechanism

Pool

Side ShowBob

( ) g

2sinvd

2θ

=( )2g

sinvh

22θ

=

Velocity mechanism boundary diagram

Slide 10

SupportInner barrel

wall

Spring constant

Weight

Frictioncoefficient

Distancecompressed

v = f(spring constant,friction coefficient,weight, distancecompressed)

Rollers Plunger Spring

Compressionlever

Compressiongauge

Side ShowBob

Velocity (m/s)75.5 – 79.2

Spring boundary diagram

Slide 11

Spring constant

Wire diameter

Free length

Number of active windings

Young’s modulus

Force =f(Young’s modulus, wire diameter, free length, number of active windings, Poisson ratio, outer diameter)

Poisson ratio

Outer diameter

Bogey test

• The duration for a Bogey test is equal to the reliability requirements being demonstrated.• For example, if a test is designed to demonstrate that a component has specific reliability at

100 hours means the test duration is100 hours, then the test is designated as a Bogey test.

• Example 1: If 95% reliability is required at 200,000 kilometres of service, then the units being tested will be removed from testing when they fail or when they complete the equivalent of 200,000 kilometres of testing. The sample size required to demonstrate reliability of r with a confidence level of c is:

Bogey: a numerical standard of performance set up as a mark to be aimed at especially in competition.

ln(r)

c)ln(1N

−=

Slide 12

Calculation of test sample size

Reliability Confidence Sample Size

99% 95% 299

99% 90% 229

99% 50% 69

95% 95% 59

95% 90% 45

95% 80% 31

90% 90% 22

90% 80% 16

Example 2: A windshield wiper motor must demonstrate 99% reliability with 90% confidence at 2 million cycles of operation. How many motors must be tested to 2 million cycles with no failures to meet these requirements:

Following table shows relationship between sample size, confidence interval & reliability

229ln(r)

c)ln(1N =

−=

Slide 13

Do tested parts represent the population?

• Variation from multiple production operators

• Variation from multiple lots of raw materials

• Variation from tool wear

• Variation from machine maintenance

• Variation from seasonal climatic changes

• Variation from supplier changes

Slide 14

Does the test represent actual use?

• Number of cycles

• Environment

• Variability in the part itself

Slide 15

What are the design requirements

• Application environment is harsh and highly variable• Vehicles must operate reliably in artic conditions and in desert conditions

• Driving profiles range from the 16 year-old male to the 90 year-old female

• An airliner may fly long haul ocean routes for 20 years

• Identical model flies short-range routes resulting in many more take-offs

• Combining this variety into a realistic test is difficult• Consider specific tests aimed at particular failure causes or failure modes

• Ensure component requirements are properly linked to the system requirements

Slide 16

95th percentile customer

• 95th percentile of what?• Cycles

• Cold temperature

• Hot temperature

• Salt

• Automobile engine• Starts

• Run time

Slide 17

Percentile vs. cycles

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0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Percentile

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ke A

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(Th

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ds)

Slide 18

Randomise loads

Slide 19

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Deceleration (g-force)

Pro

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Average Customer

95th Percentile Customer

Bayesian Weibull

• Bogey test is inefficient. By extending the test duration beyond the required life, the total time on test can often be reduced. When the test duration is extended, it is necessary to make assumptions concerning the shape of distribution of the time to fail.

• In Bayesian testing this is done by assuming a Weibull distribution for time to fail and by assuming a shape parameter

• Assume β is known:• β is dependent on the physics of failure• Year to year re-qualification

• Programme to programme

• Old design to new design

• Time raised to β is an exponential random variable• Reduces testing requirements

Slide 20

Effect of shape parameter on Time-to-FailWeibull slope β (shape) = 3.6

4.03.63.22.82.42.01.61.20.80.40.0

400

300

200

100

0

Bogeys

Fre

qu

en

cy

Shape 3.6

Scale 2

N 2500

Weibull

Histogram of Time-to-Fail

Slide 21

Effect of shape parameter on Time-to-FailWeibull slope β (shape) = 1

80706050403020100

900

800

700

600

500

400

300

200

100

0

Bogeys

Fre

qu

en

cy

Mean 15

N 2500

Histogram of Time-to-FailWeibull Shape Parameter = 1

Slide 22

Effect of shape parameter on Time-to-FailWeibull slope β (shape) = 1.8

14.212.811.29.68.06.44.83.21.60.0

350

300

250

200

150

100

50

0

Bogeys

Fre

qu

en

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Shape 1.8

Scale 5

N 2500

Weibull

Histogram of Time-to-Fail

Slide 23

Effect of shape parameter on Time-to-FailWeibull slope β (shape) = 8

1.91.71.51.31.10.90.70.5

600

500

400

300

200

100

0

Bogeys

Fre

qu

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Shape 8

Scale 1.4

N 2500

Weibull

Histogram of Time-to-Fail

Slide 24

Bayesian test design• Tests are designed to demonstrate a specific reliability at a specific

time (R95/C90 at 100,000 miles).

• To have 95% reliability at 100,000 miles the mean must be• greater than 1,500,000 miles if b = 1.0

• greater than 330,000 miles if b = 1.8

• greater than 220,000 miles if b = 3.6

• greater than 140,000 miles if b = 8.0

Slide 25

Bayesian test design

• Supported and encouraged by Automotive Industry.

• Bogey Testing is the most inefficient method of testing.• If 95% reliability is required with 90% confidence at 100 hours, there is no less efficient

method than designing a test for a duration of100 hours.

• Use extended testing.

Slide 26

Sample size to achieve R95/C90 at 1 Bogeyb: shape parameter 1 Bogey 2 Bogeys 3 Bogeys

1.0 45 22 15

1.2 45 20 12

1.5 45 16 9

2.0 45 11 5

3.5 45 4 1

7.0 45 1 1

Slide 27

Sample size to achieve R90/C90 at 1 Bogeyb: shape parameter 1 Bogey 2 Bogeys 3 Bogeys

1.0 22 11 8

1.2 22 10 6

1.5 22 8 5

2.0 22 6 3

3.5 22 2 1

7.0 22 1 1

Slide 28

Establishing the shape parameter (b)

• Can be based on physics of failure

• Should be based on (at least) 7 failures

• Can also be based on experience

• There should be a policy to build an internal database

Slide 29

Other benefits

• Failure modes will be known

• Testing can be accelerated

• Designs can be compared

• Component failures can be catalogued

• Degradation testing may be possible

• Cost savings because we understand the limits of the design

Slide 30

Example test duration

• R95/C90 at 1.5 million miles means

• to have 95% reliability at 100,000 (i.e. L5 = 1.2 million miles), with 90% confidence.

• If the Weibull slope b = 7.04 and the sample size = 8. What is the test duration?

• If 1 unit fails at time = 1.82 million miles does the product fail to demonstrate the required reliability?

Slide 31

Example test duration

Slide 32

• One unit fails after 1.82 million cycles• How long do the remaining 7 units have to survive to meet the original

test requirements?

Example

• Assume: The requirements are not met.

• Remember that the width of the confidence limits decrease as the sample size and test duration increase.

• You must make a decision:• Does the product fail to meet the requirements with this small sample size, or

• Is the product not durable enough

Slide 33

Example

• There are several options• Continue testing the surviving items

• Test additional items

• Re-design and re-test

• For extended test plans (beyond 1 lifetime) it is common to obtain failures• Failures beyond 1 lifetime do not necessarily indicate a poor design

<BayesianTestingTemplate.xls>

Slide 34

• Does a basketball player have a 90% free throw percentage?

• Verify free throw percentage with 85% confidence

• Acceptance test requires• 18 consecutive successful free throws

• If the player’s free throw percentage is exactly 90%• The probability of passing the test is

Probability of passing a bogey test

15.09.018

==p

• Demonstrate a reliability of 95% at 10 years with a confidence level of 90%

• System has a time to fail distribution for a system with a Weibull shape parameter of 2.5

• Four systems have to be tested for 26.3 years without failure

Probability of passing a Bayesian test

( )

( )

( ) ( ) 1.0563.0Test Passing

563.03.26

81.3295.0ln

10

4

81.32

3.26

5.2/1

5.2

==

=

=−

=

=

−

P

eR

• 45 units with 95% reliability at 100 time units• β = 2

• θ = 441.54

• To demonstrate 95% reliability with 90% confidence• 45 units surviving to 100 time units

• 11 units surviving to 202 time units

• 5 units surviving to 300 time units

• Compute the probability of passing each of the 3 tests above

Zero failure test plan problems

<ProbabilityOfPassingTest.xls>

Probability of passing a test – β = 2

0%

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100%

5 15 25 35 45 55 65 75 85

True L5 Life (Years)

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Probability of passing test is1 – confidence

Probability of passing a test

0%

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5 15 25 35 45 55 65 75 85

True L5 Life (Years)

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b = 1.5

b = 2.5

b = 3.5

b = 5

Confidence interval width

0 1 2 3 4 5 6 7

1

2

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Nu

mb

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of

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ilu

res

Confidence Interval Width (# of Bogeys)

b = 1.5

b = 2.5

• Statistically designed test can be defeated by selecting the parts in a non-random manner

• Select parts that are close to the nominal dimension

• The variability is reduced

• Positive bias is introduced because the parts close to nominal will perform better than those further than nominal

• Combination of reduced variability and positive bias greatly improves the probability of passing the test

• Select parts at worst case tolerance combinations

• The variability is reduced

• Negative bias is introduced because the worst-case parts will perform worse than those randomly selected

• Combination of reduced variability and negative bias greatly reduces the probability of passing the test

Sensitivity to a random sample

• Desire to determine free throw percentage for all male high school basketball players in Detroit

• Select the best free throw shooter from each team?• Similar to building prototype parts to nominal

• Select the worst free throw shooter from each team?• Similar to building prototype parts to worst case

• In either case a statistical test is nonsense

Sensitivity to a random sample

• Auction price for clocks along

• clock age

• number of bidders

• population is normally distributed• mean = 1327

• standard deviation = 393

• population L5 = 681

• But what if the sample is not random?

• Five oldest clocks are selected

• 90% confidence interval for L5 is 831 to 1641

• true L5 of 681 falls outside this range

• selecting the oldest clocks is similar to making 5 prototype parts with a new tool

Sensitivity to a random sample

Sensitivity to a random sample

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Price

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Age Greater Than 180

Entire Population

Test at Worst Case tolerance?

• What is worst case• Nominal may be worst case

• Launching projectile at 45°

• Can you produce worst case?Number of

CharacteristicsNumber of Tolerance

Combinations

2 4

3 8

4 16

5 32

10 1,024

20 1,048,576

50 1.13×1015

100 1.27×1030

Slide 45

0%

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20% 25% 30% 35% 40% 45% 50%

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Percentage of Tolerance Defined as Worst Case

2 Characteristics 3 Characteristics 4 Characteristics 5 Characteristics 6 Characteristics 7 Characteristics

Assumes Cpk = 1.33

Test design

• Use statistics as baseline

• Recommend sample size of 4 to 8• Select samples as close to worst case as possible

• More samples if testing is inexpensive and sources of variation can be incorporated

• Less if testing is expensive

• Always ask if calculations can be substituted

• If one or more samples fail prior to Bogey test is failed

• If all samples exceed Bogey• If one or two failures test is passed

• If more than two failures

• Compute reliability confidence limits and use an guideline

• Compute shape with confidence limits and use as guideline

Slide 48

Strategy

• Build confidence using a bottom up approach

• Short tests aimed at key failure modes or causes based on the P-diagram & FMEA

• Understand equations• Test unknown areas

• Use models – FEA, SPICE, CATIA, etc.

• Use standard, validated components

• Focus on System Interaction

Slide 49