Research Papers: Gas Turbines: Structures and Dynamics

Nonlinear Stochastic Dynamics, Chaos, and Reliability Analysis for a Single Degree of Freedom Model of a Rotor Blade

[+] Author and Article Information
Pankaj Kumar

Gas Turbine Design Department, Bharat Heavy Electricals Limited, Hyderabad-502032, India

S. Narayanan

Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600036, India

J. Eng. Gas Turbines Power 131(1), 012506 (Oct 13, 2008) (8 pages) doi:10.1115/1.2967720 History: Received April 03, 2008; Revised April 04, 2008; Published October 13, 2008

In turbomachinery, the analysis of systems subjected to stochastic or periodic excitation becomes highly complex in the presence of nonlinearities. Nonlinear rotor systems exhibit a variety of dynamic behaviors that include periodic, quasiperiodic, chaotic motion, limit cycle, jump phenomena, etc. The transitional probability density function (PDF) for the random response of nonlinear systems under white or colored noise excitation (delta-correlated) is governed by both the forward Fokker–Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the stationary and transient form of the forward FP equation corresponding to two state nonlinear systems by standard sequential finite element (FE) method using C0 shape functions and Crank–Nicholson time integration scheme. For computing the reliability of system, the transient FP equation is solved on the safe domain defined by D barriers using the FE method. A new approach for numerical implementation of path integral (PI) method based on non-Gaussian transition PDF and Gauss–Legendre scheme is developed. In this study, PI solution procedure is employed to solve the FP equation numerically to examine some features of chaotic and stochastic responses of nonlinear rotor systems.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

Joint PDF for the Van der Pol oscillator with ξ=0.1 and D=0.1

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Figure 2

Joint PDF for the Van der Pol oscillator with ξ=1.0 and D=0.1

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Figure 3

Joint PDF for the hardening Duffing oscillator: (a) t=2.827s and (b) t=31.419s

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Figure 4

Phase space safe domain for D-barriers for the first passage problem

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Figure 5

Probability of failure versus time in natural periods for the Van der Pol system: B=3, ξ=0.5, and D=1; (—) forward solution and (●) backward solution

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Figure 6

Marginal PDF of (a) displacement and (b) velocity for the white noise excited Duffing oscillator; (—) exact results, (●), FEM, (◼) PI method

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Figure 7

Marginal PDF of the (a) displacement and (b) velocity. Key as in Fig. 6.

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Figure 8

A strange attractor for the Duffing oscillator given by Eq. 43 with β=0.075, ω=1.0, and f0=0.3

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Figure 9

Invariant measure on X1: (●) noise intensity at 0.02 and (—) noise intensity at 0.07



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