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.

1.
Srinivasan
,
A. V.
, 1997, “
Flutter and Resonant Vibration Characteristics of Engine Blade
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
119
(
4
), pp.
742
775
.
2.
Griffin
,
J. H.
, and
Hoosac
,
T. M.
, 1984, “
Model Development and Statistical Investigation of Turbine Blade Mistuning
,”
ASME J. Vib., Acoust., Stress, Reliab. Des.
0739-3717,
106
, pp.
204
210
.
3.
Cha
,
D.
and
Sinha
,
A.
, 1999, “
Statistics of Response of a Mistuned Baded Disk Assembly Subjected to White Noise and Narrow Band Excitation
,”
ASME J. Eng. Gas Turbines Power
0742-4795,
119
, pp.
710
717
.
4.
,
C.
,
Green
,
J. S.
,
Vahdati
,
M.
, and
Imregun
,
M.
, 2001, “
A Non-Linear Integrated Aeroelasticity Method for the Prediction of Turbine Forced Response With Friction Damper
,”
Int. J. Mech. Sci.
0020-7403,
43
, pp.
2715
2736
.
5.
Sunderrajan
,
P.
, and
Noah
,
S. T.
, 1997, “
Dynamics of Forced Nonlinear Systems Using Shooting∕Arc-Length Continuation Method-Application to Rotor Systems
,”
ASME J. Vibr. Acoust.
0739-3717,
119
, pp.
9
20
.
6.
Sogliero
,
G.
, and
Srinivasan
,
A. V.
, 1980, “
Fatigue Life Estimates of Mistuned Blades Via a Stochastic Approach
,”
AIAA J.
0001-1452,
18
(
83
), pp.
318
323
.
7.
Roberts
,
J. B.
, and
Spanos
,
P. D.
, 1990,
Random Vibration and Statistical Linearization
,
Wiley
,
New York
.
8.
Crandall
,
S. H.
, 1963, “
Perturbation Techniques for Random Vibration of Nonlinear Systems
,”
J. Acoust. Soc. Am.
0001-4966,
35
(
11
), pp.
1700
1705
.
9.
Wojtkiewicz
,
S. F.
, and
Spencer
,
B. F.
, Jr.
, and
Bergman
,
L. A.
, 1996, “
On the Cumulant-Neglect Closure Method in Stochastic Dynamics
,”
Int. J. Non-Linear Mech.
0020-7462,
31
(
3
), pp.
657
684
.
10.
Nigam
,
N. C.
, 1983,
Introduction for Random Vibration
,
MIT
,
Cambridge, MA
.
11.
Wehner
,
M. F.
, and
Wolfer
,
W. G.
, 1983, “
Numerical Evaluation of Path-Integral Solutions to Fokker–Planck Equation
,”
Phys. Rev. A
1050-2947,
27
(
5
), pp.
2663
2670
.
12.
Naess
,
A.
, and
Moe
,
V.
, 2000, “
Efficient Path Integration Method for Nonlinear Dynamics System
,”
Probab. Eng. Mech.
0266-8920,
15
, pp.
221
231
.
13.
Yu
,
J. S.
,
Cai
,
G. Q.
, and
Lin
,
Y. K.
, 1997, “
A New Path Integration Procedure Based on Gauss–Legendre Scheme
,”
Int. J. Non-Linear Mech.
0020-7462,
32
, pp.
759
768
.
14.
Lin
,
H.
and
Yim
,
S. C. S.
, 1994, “
A Path Integral Procedure for the Analysis of a Noisy Nonlinear System
,”
Proceedings of the Second International Conference on Computational Stochastic Mechanics
,
Spanos
,
Athens
, June 12–15, pp.
371
377
.
15.
Langley
,
R. S.
, 1985, “
A Finite Element Method for the Statistics of Non-Linear Random Vibration
,”
J. Sound Vib.
0022-460X,
101
(
1
), pp.
41
54
.
16.
Spencer
,
B. F.
, and
Bergman
,
L. A.
, 1991, “
Numerical Solution of the Fokker–Planck Equation for First Passage Probability
,”
Computational Stochastic Mechanics
,
P. D.
Spanos
and
C. A.
Brebbia
, eds.,
Elsevier
,
Southampton
, pp.
359
370
.
17.
Kumar
,
P.
, and
Narayanan
,
S.
, 2006, “
Solution of Fokker–Planck Equation by Finite Element and Finite Difference Methods for Nonlinear System
,”
0256-2499,
31
(
4
), pp.
455
473
.
18.
Wojtkiewicz
,
S. F.
,
Bergman
,
L. A.
, and
Spencer
,
B. F.
, Jr.
, 1994, “
Robust Numerical Solution of the Fokker–Planck–Kolmogorov Equation for Two Dimensional Stochastic Dynamical Systems
,” Department of Aeronautical and Astronautical Engineering,
University of Illinois at Urbana-Champaign
, Technical Report No. AAE 94-08.
19.
Kunert
,
A.
, 1991, “
Efficient Numerical Solution of Multidimensional Fokker–Planck Equations With Chaotic and Nonlinear Random Vibration
,”
Vibration Analysis: Analytical and Computational
,
T. C.
Huang
, eds., Vol.
DE-37
, pp.
57
60
.
20.
Soong
,
T.
, and
Grigoriu
,
M.
, 1993,
Random Vibration of Mechanical and Structural Systems
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
21.
Wong
,
E.
, and
Zakai
,
M.
, 1965, “
On the Relation Between Ordinary and Stochastic Differential Equation
,”
Int. J. Eng. Sci.
0020-7225,
3
, pp.
213
229
.
22.
Risken
,
H.
, 1989,
The Fokker–Planck Equation: Methods of Solution and Applications
,
Springer-Verlag
,
New York
.
23.
Crandall
,
S. H.
, 1970, “
First Crossing Probabilities of the Linear Oscillator
,”
J. Sound Vib.
0022-460X,
12
(
3
), pp.
285
299
.
24.
Zhang
,
D. S.
,
Wei
,
G. W.
,
Kouri
,
D. J.
, and
Hoffman
,
D. K.
, 1997, “
Numerical Method for the Nonlinear Fokker–Planck Equation
,”
Phys. Rev. E
1063-651X,
56
(
1
), pp.
1197
1206
.
25.
Naess
,
A.
, 2000, “
Chaos and Nonlinear Stochastic Dynamics
,”
Probab. Eng. Mech.
0266-8920,
15
, pp.
37
47
.
26.
Jung
,
P.
, and
Hänggi
,
P.
, 1990, “
Invariant Measure of a Driven Nonlinear Oscillator With External Noise
,”
Phys. Rev. Lett.
0031-9007,
65
, pp.
3365
3368
.