Design for Lifecycle Cost using Time-Dependent Reliability

Design for Lifecycle Cost using Time-Dependent Reliability Amandeep Singh Zissimos P. Mourelatos Jing Li Mechanical Engineering Department Oakland University Rochester, MI 48309, USA : Distribution Statement A. Approved for public release 1

Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 13 MAY 2009 2. REPORT TYPE Briefing Charts 3. DATES COVERED 13-05-2009 to 13-05-2009 4. TITLE AND SUBTITLE DESIGN FOR LIFECYCLE COST USING TIME-DEPENDENT RELIABILITY 6. AUTHOR(S) Amandeep Singh; Zissimos Mourelatos; Jing Li 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Oakland University,Mechanical Engineering Department,Rochester,MI,48309 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) U.S. Army TARDEC, 6501 E.11 Mile Rd, Warren, MI, 48397-5000 8. PERFORMING ORGANIZATION REPORT NUMBER ; #22498 10. SPONSOR/MONITOR S ACRONYM(S) TARDEC 11. SPONSOR/MONITOR S REPORT NUMBER(S) #22498 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT N/A 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Same as Report (SAR) 18. NUMBER OF PAGES 33 19a. NAME OF RESPONSIBLE PERSON Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

Stress SPL (t) (t) ARC Problem Definition Time t Random Field Time t Response(t) = f [ E(t), Degradation/Wear(t), Load(t) ] Random Process approach to reliability-based design is needed time-dependent reliability 2

Frequency Frequency Frequency Frequency ARC Input variables Problem Definition System Responses System Reliability Time t L t L Time System Reliability Variable #1 Response #1 Quality Time Time t L t L Time Quality = Reliability (t = 0) Variable #n Response #m 3

System Reliability ARC What can we get from Time- Dependent Reliability? Design for: Lifecycle cost Quality Warranty Maintenance schedule for CBM R(t) Time for Maintenance Acceptable Reliability Time t 4

Design for Lifecycle Cost Lifecycle Cost = Production Cost + Inspection Cost + Expected Variable Cost Quality Time-Dependent System Reliability Accurate and efficient predictive tools are, therefore, needed to estimate Time-dependent System Reliability. 5

C L Design for Lifecycle Cost E d, X, t, r C d, XC d, X, t C,, t, r 0 d X f P I V f Lifecycle Cost Production Cost Inspection Cost Expected Variable Cost F c T C E V t Pt, t, such that gxt L Final time t t r rt c d, X, f, cf t e ft t dt 0, t L f 0 Cost of failure at time t Interest rate PDF of time to failure time 0 6

min d, μ s. t. X, σ X F C i Q i Problem Statement: Design for Lifecycle Cost d, μ, t r L x f, d X, t p ( ), 0 f t0 System Reliability Quality F c R d X, t p ( t ), k f k t k t f d μ X d μ μ L d U X L X U Quantification of time-dependent reliability is a major challenge in this research. 7

Definitions / Observations Reliability: Ability of a system to carry out a function in a time period [0, t L ] p c f c t F t P t L T L Prob. of Time to Failure F c T t Pt, t, such that gxt L 0, t L 0 Cumulative Prob. of Failure F i T t L P g X t L, t 0 L Instantaneous Prob. of Failure Time-Invariant Reliability 0 tl F i t T L time Time-Variant Reliability 0 tl time c F T tl 8

Calculation of Cumulative Probability of Failure State-of-the-art Approaches PHI2 method (Andrieu-Renaud, et al., RESS, 2004) Set-Based approach (Son and Savage, Quality & Rel. Engin., 2007) t time t1 t 2 tk 2 t K 1 0 t F Kt State-of-the-art approaches are in general, inaccurate due to: Choice of t Not including contribution of all discrete times t F 9

Cumulative Probability of Failure Composite Limit State t K MPP1 Safe region MPP2 t 0 t0 Unsafe region t K Example 1: Linear Limit State g X1, X 2, t X1 tx 2 2 X ~ N 5, t 1 1 0 Composite limit state Single MPP of instantaneous limit states evolves into multiple MPPs of composite limit state. F c T t X 2 2 ~ N 0, 1 t K t P gxt F K k0 k, t k 1/31/2012 10

Cumulative Probability of Failure Composite Limit State Example 2: g X1 1000X 2 sin 4t X 2 X1, X 2 1, 12000 52966 2 X1 ~ N4.58E08, 5E07 X 2 2 0. 7 ~ N0, 11

Composite Limit State: Example 2 t = 0 12

Composite Limit State: Example 2 t = 0 t = 0.125 13

Composite Limit State: Example 2 t = 0 t = 0.125 t = 0.25 14

Composite Limit State: Example 2 t = 0.375 t = 0.125 t = 0 t = 0.25 15

Composite Limit State: Example 2 t = 0.5 t = 0.375 t = 0.125 t = 0 t = 0.25 16

Composite Limit State: Example 2 17

Calculation of Probability of Failure Two Proposed Approaches Reliability Index Approach Limit State is kept Time-dependent i.e. g( d, Χ, t) 0 Maximum Response Approach Limit State is converted into Time-Independent i.e g( d, Χ) 0 18

Calculation of Probability of Failure Reliability Index Approach: t K t 0 s.t. min U 2 t g U, t U, t 0 t 0 t max MPP1 Safe region MPP2 Unsafe region Time is treated as an additional design variable in RIA optimization. t0 t K 1/31/2012 19

y(x,t) ARC Cumulative Probability of Failure Maximum Response Approach: t y 0 t 1 t 2 tk 2 t K 1 t F time time y max ( d, X) t min max t t max y( d, X, t) F c T t F P( y max ( ) y t ) P( y t y max ( ) 0) Composite Limit-State as time-independent is defined as: g t d, X y y max 0 20

Calculation of p f : Two-DOF System 21

Two-DOF System 2 m, m, m 55 Kg, m Kg 2,, 33E04 N m mc ~ N 5 ks ~ N k k k / t k 3E04 N / u : unit impulse; 0 t 5s m 22

Two-DOF System- Multiple MPPs T=0.2 sec T=1 sec Maximum Response method Niching GA optimization to search for multiple MPPs 23

Two-DOF System- Comparison of Pf 24

Design of a Roller Clutch using Lifecycle Cost 25

Roller Clutch Random Design Variables: D: Hub diameter, mm d: Roller diameter, mm A: Cage inner diameter, mm D, d, and A are normally distributed Due to degradation: D D 1 kt d d 1 kt A A 1 kt with: k 2.5E 04 mm/ year 26

Constraints: Roller Clutch Contact angle 0.11 0. 06 rad 0.05 4 Limit States cos 1 Torque 3000 Hoop stress D A d d 0.17 g g h 400 g g Nm MPa 1 D, d, A 0.05 cos 0 1 D d A d 1 D, d, A cos 0.17 0 2 N 2 c1 Dd ( ) D d D d A d 2 2 D d 3 c 2 D, d, A 3000 NL 1 S 0 c 1 2 2 2 S B A 4, 2 2 c D d, A 400E06 0 A B 4( D d) A 27

s. t. ARC Minimize Lifecycle Cost Case 1 F i μ X, σx, t0 Roller Clutch: Problem Statement min μ X 0, σ X C μ, σ, t, L x X f 4 P( ( gi ( D, d, A, t ) 0)) p f 0 i r σ μ X X L X U σ μ t 0 0. 0013 0 Case 2 F i 4 μ, t 0 P( ( gi ( D, d, A, t0) 0)) p f t0 0 0. 0013 F c X, σx 0 μ, σ X, t X 7. 5 P( i 4 i ( g i ( D, d, A, t) 0)) p f t 7.5 0. 005 X X σ μ L X U Case 3 F c μ X, σ, t X 10 P( 4 i ( g i ( D, d, A, t) 0)) p f t 10 0. 0716 28

Roller Clutch: Problem Statement where: Total Cost, C L C P C I C E V C P 0.75 0.65 0.88 3.5 3.0 0.5 3 D 3 d 3 A C I Scrap cost/unit 20FQ ( X, t0) C E V t f 0 20e rt f c R t dt Failure cost/unit (warranty cost) t f 10 years r 3% 29

Roller Clutch: Results Initial Design vs. Case 1 Case 1 vs. Case 2 and Case 3 Initial Optimal Design Design Case 1 Case 2 Case 3 Objective Total Cost 28.2275 23.876 24.5440 21.1896 Production Cost 17.3900 21.3340 23.4446 19.9383 Inspection Cost 0.7677 0.0260 0.0260 0.6596 Expected Variable Cost 10.0697 2.5161 1.07340 0.5918 30

Summary/Conclusions A new method to calculate the Cumulative Probability of failure is presented for linear and non-linear problems. The design study of the roller clutch showed that: Lifecycle cost can be reduced by controlling the probability of failure though time. Higher lifecycle cost due to higher initial quality does not guarantee acceptable reliability. 31

Challenges/Future Work Work Improve further efficiency by: Random process characterization using timeseries modeling techniques. Solving RBDO problem using Probabilistic Re- Analysis which uses a single MCS Apply presented ideas/approaches to the Army related problems 32

Q & A **Disclaimer: Reference herein to any specific commercial company, product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the Department of the Army (DoA). The opinions of the authors expressed herein do not necessarily state or reflect those of the United States Government or the DoA, and shall not be used for advertising or product endorsement purposes.** 33