Research Papers

Probability-Based Prediction of Degrading Dynamic Systems

[+] Author and Article Information
Gordon J. Savage

e-mail: gjsavage@uwaterloo.ca

Turuna S. Seecharan

e-mail: tseechar@uwaterloo.ca
Department of Systems Design Engineering,
University of Waterloo,
Waterloo, ON, N2L 3G1, Canada

Young Kap Son

Department of Mechanical & Automotive Engineering,
Andong National University,
Andong-si, Gyeongsanbuk-do 760-749,
South Korea
e-mail: ykson@andong.ac.kr

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received April 2, 2012; final manuscript received November 30, 2012; published online January 17, 2013. Assoc. Editor: Zissimos P. Mourelatos.

J. Mech. Des 135(3), 031002 (Jan 17, 2013) (14 pages) Paper No: MD-12-1184; doi: 10.1115/1.4023280 History: Received April 02, 2012; Revised November 30, 2012

This paper presents a methodology to provide the cumulative failure distribution (CDF) for degrading, uncertain, and dynamic systems. The uniqueness and novelty of the methodology is that long service time over which degradation occurs has been augmented with much shorter cycle time over which there is uncertainty in the system dynamics due to uncertain design variables. The significance of the proposed methodology is that it sets the foundation for setting realistic life-cycle management policies for dynamic systems. The methodology first replaces the implicit mechanistic model with a simple explicit meta-model with the help of design of experiments and singular value decomposition, then transforms the dynamic, time variant, probabilistic problem into a sequence of time invariant steady-state probability problems using cycle-time performance measures and discrete service time, and finally, builds the CDF as the summation of the incremental service-time failure probabilities over the planned life time. For multiple failure modes and multiple discrete service times, set theory establishes a sequence of true incremental failure regions. A practical implementation of the theory requires only two contiguous service-times. Probabilities may be evaluated by any convenient method, such as Monte Carlo and the first-order reliability method. Error analysis provides ways to control errors with regards to probability calculations and meta-model fitting. A case study of a common servo-control mechanism shows that the new methodology is sufficiently fast for design purposes and sufficiently accurate for engineering applications.

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Fig. 1

An uncertain dynamic response and the effects of degradation: (a) t = t0 and (b) t = tl

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Fig. 2

Limit-state surface with failure side shaded

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Fig. 3

Trajectories of a limit-state surface over time

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Fig. 4

Time-variant limit-state surfaces: (a) arbitrary trajectories and (b) greatly correlated trajectories

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Fig. 5

Schematic of a position-control system

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Fig. 6

Angular speed and errors: (a) four comparable responses for nominal design values and (b) the error estimate over cycle time for SVD with repsect to the mechanistic model and RSM and Kriging with respect to the mechanistic model

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Fig. 7

Percentage error for each method over time

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Fig. 8

The CDFs estimated using RSM, Kriging, and the mechanistic model with normal variables when μCR = 0.004

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Fig. 9

The CDFs estimated using RSM, Kriging, and the mechanistic model with log-normal variables when μCR = 0.004

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Fig. 10

Differences between the estimated mechanistic CDF and the estimated CDFs from RSM and Kriging over service time for normal variables and μCR = 0.004

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Fig. 11

Differences between the estimated mechanistic model CDF and the estimated CDFs from RSM and Kriging over service time for log-normal variables and μCR = 0.004

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Fig. 12

The CDFs estimated using RSM, Kriging and the mechanistic model with normal variables when μCR = 0.008

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Fig. 13

The CDF estimated using RSM, Kriging, and the mechanistic model with log-normal variables when μCR = 0.008

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Fig. 14

Differences between the estimated mechanistic model CDF and the estimated CDFs from RSM and Kriging over service time for normal variables and μCR = 0.008

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Fig. 15

Differences between the estimated mechanistic model CDF and the estimated CDFs from RSM and Kriging over service time for log-normal variables and μCR = 0.008

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Fig. 16

The limit-state surfaces plotted using the meta-models at three service times




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