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.

Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.


Soong, T. T. , and Grigoriu, M. , 1993, Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, NJ.
Roberts, J. B. , and Spanos, P. D. , 1999, Random Vibration and Statistical Linearization, Dover Publications, Mineola, NY.
Li, C.-C. , and Kiureghian, A. D. , 1993, “ Optimal Discretization of Random Fields,” J. Eng. Mech., 119(6), pp. 1136–1154. [CrossRef]
Zhang, J. , and Ellingwood, B. , 1994, “ Orthogonal Series Expansions of Random Fields in Reliability Analysis,” J. Eng. Mech., 120(12), pp. 2660–2677. [CrossRef]
Sudret, B. , and Der Kiureghian, A. , 2000, “ Stochastic Finite Element Methods and Reliability—A State of the Art Report,” University of California, Berkeley, CA, Report No. UCB/SEMM-2000/08. http://citeseerx.ist.psu.edu/viewdoc/download?doi=
Spanos, P. D. , Kougioumtzoglou, I. A. , and Soize, C. , 2011, “ On the Determination of the Power Spectrum of Randomly Excited Oscillators Via Stochastic Averaging: An Alternative Approach,” Probab. Eng. Mech., 26(1), pp. 10–15. [CrossRef]
Spanos, P. D. , and Kougioumtzoglou, I. A. , 2012, “ Harmonic Wavelets Based Statistical Linearization for Response Evolutionary Power Spectrum Determination,” Probab. Eng. Mech., 27(1), pp. 57–68. [CrossRef]
Shields, M. D. , Deodatis, G. , and Bocchini, P. , 2011, “ A Simple and Efficient Methodology to Approximate a General Non-Gaussian Stationary Stochastic Process by a Translation Process,” Probab. Eng. Mech., 26(4), pp. 511–519. [CrossRef]
Joo, H. K. , and Sapsis, T. , 2016, “ A Moment-Equation-Copula-Closure Method for Nonlinear Vibrational Systems Subjected to Correlated Noise,” Probab. Eng. Mech., 46, pp. 120–132. [CrossRef]
Mourelatos, Z. P. , Majcher, M. , and Geroulas, V. , 2016, “ Time-Dependent Reliability Analysis of Vibratory Systems With Random Parameters,” ASME J. Vib. Acoust., 138(3), p. 031007. [CrossRef]
Tsianika, V. , Geroulas, V. , Mourelatos , and Baseski, I. , 2017, “ A Methodology for Fatigue Life Estimation of Linear Vibratory Systems Under Non-Gaussian Loads,” SAE Paper No. 2017-01-0197.
Melchers, R. E. , 1999, Structural Reliability Analysis and Prediction, 2nd ed., Wiley, Chichester, UK.
Hu, Z. , Li, H. , Du, X. , and Chandrashekhara, K. , 2013, “ Simulation-Based Time-Dependent Reliability Analysis for Composite Hydrokinetic Turbine Blades,” Struct. Multidiscip. Optim., 47(5), pp. 765–781. [CrossRef]
Hu, Z. , and Du, X. , 2013, “ Time-Dependent Reliability Analysis With Joint Upcrossing Rates,” Struct. Multidiscip. Optim., 48(5), pp. 893–907. [CrossRef]
Wang, Z. , and Wang, P. , 2012, “ A Nested Extreme Response Surface Approach for Time-Dependent Reliability-Based Design Optimization,” ASME J. Mech. Des., 134(12), p. 121007. [CrossRef]
Andrieu-Renaud, C. , Sudret, B. , and Lemaire, M. , 2004, “ The PHI2 Method: A Way to Compute Time-Variant Reliability,” Reliab. Eng. Saf. Syst., 84(1), pp. 75–86. [CrossRef]
Madsen, P. H. , and Krenk, S. , 1984, “ An Integral Equation Method for the First Passage Problem in Random Vibration,” ASME J. Appl. Mech., 51(3), pp. 674–679. [CrossRef]
Rice, S. O. , 1944, “ Mathematical Analysis of Random Noise,” Bell Syst. Tech. J., 23(3), pp. 282–332. [CrossRef]
Rackwitz, R. , 1998, “ Computational Techniques in Stationary and Non-Stationary Load Combination—A Review and Some Extensions,” J. Struct. Eng., 25(1), pp. 1–20. https://www.researchgate.net/publication/279895900_Computational_techniques_in_stationary_and_non-stationary_load_combination_-_A_review_and_some_extensions
Hu, Z. , and Du, X. , 2012, “ Reliability Analysis for Hydrokinetic Turbine Blades,” Renewable Energy, 48, pp. 251–262. [CrossRef]
Mourelatos, Z. P. , Majcher, M. , Pandey, V. , and Baseski, I. , 2015, “ Time-Dependent Reliability Analysis Using the Total Probability Theorem,” ASME J. Mech. Des., 137(3), p. 031405. [CrossRef]
Shinozuka, M. , and Jan, C. , 1972, “ Digital Simulation of Random Processes and Its Applications,” J. Sound Vib., 25(1), pp. 111–128. [CrossRef]
Yamazaki, Y. , and Shinozuka, M. , 1988, “ Digital Generation of Non-Gaussian Stochastic Fields,” J. Eng. Mech., 114(7), pp. 1183–1197. [CrossRef]
Sakamoto, S. , and Ghanem, R. , 2002, “ Simulation of Multi-Dimensional Non-Gaussian Non-Stationary Random Fields,” Probab. Eng. Mech., 17(2), pp. 167–176. [CrossRef]
Xiu, D. , and Karniadakis, G. , 2002, “ The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations,” SIAM J. Sci. Comput., 24(2), pp. 619–644. [CrossRef]
Dick, J. , Kuo, F. Y. , and Sloan, I. H. , 2013, “ High Dimensional Integration—The Quasi Monte Carlo Way,” Acta Numerica, 22, pp. 133–288. [CrossRef]
Ye, Q. , Li, W. , and Sudjianto, A. , 2000, “ Algorithmic Construction of Optimal Symmetric Latin Hypercube Designs,” J. Stat. Plann. Inference, 90(1), pp. 145–159. [CrossRef]
Newland, D. E. , 1993, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, 3rd ed., Dover Publications, Mineola, NY.


Grahic Jump Location
Fig. 1

Schematic of input–output notation

Grahic Jump Location
Fig. 2

Schematic of Duffing oscillator

Grahic Jump Location
Fig. 3

Autocorrelation function and PSD of input F(t)

Grahic Jump Location
Fig. 4

Eigenvalues of Cξξ(ti,tj) for different tS

Grahic Jump Location
Fig. 5

Input trajectories for 0≤t≤tS

Grahic Jump Location
Fig. 6

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

Grahic Jump Location
Fig. 7

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

Grahic Jump Location
Fig. 8

Marginal PDF of output X(t)

Grahic Jump Location
Fig. 9

Autocorrelation function of output X(t)

Grahic Jump Location
Fig. 10

Trajectories of X(t) using its stochastic metamodel

Grahic Jump Location
Fig. 11

Time-dependent probability of failure for large α values

Grahic Jump Location
Fig. 12

Time-dependent probability of failure for small α values

Grahic Jump Location
Fig. 13

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

Grahic Jump Location
Fig. 14

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




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In