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Research Papers

Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters

[+] Author and Article Information
Zissimos P. Mourelatos

Mechanical Engineering Department,
Oakland University,
2200 N. Squirrel Road,
Rochester, MI 48309
e-mail: mourelat@oakland.edu

Monica Majcher

Mechanical Engineering Department,
Oakland University,
2200 N. Squirrel Road,
Rochester, MI 48309
e-mail: mtmajch2@oakland.edu

Vasileios Geroulas

Mechanical Engineering Department,
Oakland University,
2200 N. Squirrel Road,
Rochester, MI 48309
e-mail: vgeroula@oakland.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received March 29, 2015; final manuscript received January 21, 2016; published online April 7, 2016. Assoc. Editor: Mohammed Daqaq.

J. Vib. Acoust 138(3), 031007 (Apr 07, 2016) (9 pages) Paper No: VIB-15-1102; doi: 10.1115/1.4032720 History: Received March 29, 2015; Revised January 21, 2016

The field of random vibrations of large-scale systems with millions of degrees-of-freedom (DOF) is of significant importance in many engineering disciplines. In this paper, we propose a method to calculate the time-dependent reliability of linear vibratory systems with random parameters excited by nonstationary Gaussian processes. The approach combines principles of random vibrations, the total probability theorem, and recent advances in time-dependent reliability using an integral equation involving the upcrossing and joint upcrossing rates. A space-filling design, such as optimal symmetric Latin hypercube (OSLH) sampling, is first used to sample the input parameter space. For each design point, the corresponding conditional time-dependent probability of failure is calculated efficiently using random vibrations principles to obtain the statistics of the output process and an efficient numerical estimation of the upcrossing and joint upcrossing rates. A time-dependent metamodel is then created between the input parameters and the output conditional probabilities allowing us to estimate the conditional probabilities for any set of input parameters. The total probability theorem is finally applied to calculate the time-dependent probability of failure. The proposed method is demonstrated using a vibratory beam example.

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Figures

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Fig. 1

Schematic of input–output notation

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Fig. 2

Notation for v+(τ) and v++(t,τ) calculation

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Fig. 3

Impulse response function h(t)

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Fig. 4

Beam under random loading [28]

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Fig. 5

Input force spectrum

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Fig. 6

Correlation coefficient of input force

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Fig. 7

Threshold and output mean functions for beam example

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Fig. 8

Comparison of covariance function CYY(t) with MCS for b=0.015 and h=0.04

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Fig. 9

Correlation coefficient function for b=0.015 and h=0.04

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Fig. 10

Instantaneous reliability index (upper truncated curves) and upcrossing rate (lower curves) for b=0.015 and h=0.04

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Fig. 11

Comparison of conditional time-dependent probability of failure  P(E/X) (solid line) with MCS (dotted line) for five design points

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Fig. 12

Design of 45 OSLH points

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Fig. 13

 P(E/X) curves for the 45 OSLH points

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Fig. 14

Comparison of conditional probability of failure for mean values of b and h

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Fig. 15

Comparison of time-dependent probability of failure with MCS

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