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Research Papers

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

[+] Author and Article Information
Pol D. Spanos

Honorary Mem. ASME
L.B. Ryon Chair in Engineering
Department of Mechanical Engineering
and Materials Science,
Rice University,
6100 Main Street,
Houston, TX 77005-1827
e-mail: spanos@rice.edu

Alberto Di Matteo

Dipartimento di Ingegneria Civile,
Ambientale, Aerospaziale, dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy
e-mail: alberto.dimatteo@unipa.it

Yezeng Cheng

Department of Mechanical Engineering
and Materials Science,
Rice University,
6100 Main Street,
Houston, TX 77005-1827
e-mail: yc22@rice.edu

Antonina Pirrotta

Dipartimento di Ingegneria Civile,
Ambientale, Aerospaziale, dei Materiali (DICAM),
Università degli Studi di Palermo,
Viale delle Scienze,
Palermo 90128, Italy
e-mail: antonina.pirrotta@unipa.it

Jie Li

School of Civil Engineering,
Tongji University,
1239 Siping Road,
Shanghai 200092, China
e-mail: lijie@tongji.edu.cn

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 8, 2016; final manuscript received August 8, 2016; published online September 14, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(12), 121003 (Sep 14, 2016) (9 pages) Paper No: JAM-16-1282; doi: 10.1115/1.4034460 History: Received June 08, 2016; Revised August 08, 2016

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

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Figures

Grahic Jump Location
Fig. 1

Galerkin scheme vis-à-vis MCS results (20,000 samples) for a linear oscillator (B=1, α=0.6, Cα=0.05) for various values of N: (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 2

Galerkin scheme vis-à-vis MCS results (20,000 samples) for a linear oscillator (α=0.6, Cα=0.05) for various levels of the barrier B: (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 3

Galerkin scheme vis-à-vis MCS results (20,000 samples) for a linear oscillator (B=1, α=0.6) for various levels of Cα: (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 4

Galerkin scheme vis-à-vis MCS results (20,000 samples) for a linear oscillator (B=1, Cα=0.05) for various levels of α: (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 5

Galerkin scheme vis-à-vis MCS results (20,000 samples) for the nonlinear Van der Pol oscillator (Cα=−0.05, α=0.6, ε=0.2): (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 6

Galerkin scheme vis-à-vis MCS results (20,000 samples) for the nonlinear Van der Pol oscillator (Cα=−0.05, α=0.6, ε=1): (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 7

Galerkin scheme vis-à-vis MCS results (20,000 samples) for the Duffing oscillator (Cα=0.05, α=0.6, ε=0.2): (a) survival probability and (b) first-passage probability density function

Grahic Jump Location
Fig. 8

Galerkin scheme vis-à-vis MCS results (20,000 samples) for the Duffing oscillator (Cα=0.05, α=0.6, ε=1): (a) survival probability and (b) first-passage probability density function

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