Research Papers: Design Automation

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads

[+] Author and Article Information
Vasileios Geroulas

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

Zissimos P. Mourelatos

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

Vasiliki Tsianika

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

Igor Baseski

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

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 25, 2017; final manuscript received September 21, 2017; published online December 14, 2017. Assoc. Editor: Xiaoping Du.

J. Mech. Des 140(2), 021404 (Dec 14, 2017) (9 pages) Paper No: MD-17-1368; doi: 10.1115/1.4038212 History: Received May 25, 2017; Revised September 21, 2017

A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on polynomial chaos expansion (PCE), Karhunen–Loeve (KL) expansion, and quasi Monte Carlo (QMC). The latter is used to estimate multidimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness, and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion (EOM) are time integrated for each of the M points, and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE–KL–QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

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Fig. 1

Schematic of input–output notation

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Fig. 2

Schematic of Duffing oscillator

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Fig. 3

Autocorrelation function and PSD of input F(t)

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Fig. 13

Time-dependent probability of failure for large α values—random coefficient case

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Fig. 14

Time-dependent probability of failure for small α values—random coefficient case

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Fig. 4

Eigenvalues of Cξξ(ti,tj) for different tS

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Fig. 5

Input trajectories for 0≤t≤tS

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Fig. 6

Marginal PDF of F(t) for different values of M

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Fig. 7

M = 200 calculated output realizations for 0≤t≤tS

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Fig. 8

Marginal PDF of output X(t)

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Fig. 9

Autocorrelation function of output X(t)

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Fig. 12

Time-dependent probability of failure for small α values

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Fig. 10

Trajectories of X(t) using its stochastic metamodel

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Fig. 11

Time-dependent probability of failure for large α values



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