This paper investigates the influences of nonzero mean Poisson impulse amplitudes on the response statistics of dynamical systems. New correction terms of the extended Itô calculus, as a generalization of the Wong–Zakai correction terms in the case of normal excitations, are adopted to consider the non-normal property in the case of Poisson process. Due to these new correction terms, the corresponding drift and diffusion coefficients of Fokker–Planck–Kolmogorov (FPK) equation have to be modified and they become more complicated. Herein, the exponential–polynomial closure (EPC) method is employed to solve such a complex FPK equation. Since there are no exact solutions, the efficiency of the EPC method is numerically evaluated by the simulation results. Three examples of different excitation patterns are considered. Numerical results indicate that the influence of nonzero mean impulse amplitudes on system responses depends on the excitation patterns. It is negligible in the case of parametric excitation on displacement. On the contrary, the influence becomes significant in the cases of external excitation and parametric excitation on velocity.

References

References
1.
Grigoriu
,
M.
,
1995
, “
Equivalent Linearization for Poisson White Noise Input
,”
Probab. Eng. Mech.
,
10
(
1
), pp.
45
51
.
2.
Yue
,
X.
,
Xu
,
Y.
, and
Yuan
,
J.
,
2013
, “
Approximate Stationary Solution for Beam-Beam Interaction Model With Parametric Poisson White Noise
,”
CMES-Comput. Model. Eng.
,
93
(4), pp.
277
291
.
3.
Wang
,
J. P.
, and
Chang
,
S. C.
,
2015
, “
Evidence in Support of Seismic Hazard Following Poisson Distribution
,”
Phys. A
,
424
, pp.
207
216
.
4.
Roberts
,
J. B.
,
1972
, “
System Response Random Impulses
,”
J. Sound Vib.
,
24
(
1
), pp.
23
34
.
5.
Iwankiewicz
,
R.
,
Nielsen
,
S. R. K.
, and
Christensen
,
P. T.
,
1990
, “
Dynamic Response of Nonlinear Systems to Poisson Distributed Pulse Train: Markov Approach
,”
Nonlinear Structural Systems Under Random Condition
,
F.
Casciati
,
I.
Elishakoff
, and
J. B.
Roberts
, eds., Elsevier, Amsterdam, The Netherlands, pp.
223
238
.
6.
Lutes
,
L. D.
, and
Hu
,
S.-L. J.
,
1986
, “
Non-Normal Stochastic Response of Linear Systems
,”
J. Eng. Mech.
,
112
(2), pp.
127
141
.
7.
Proppe
,
C.
,
2002
, “
The Wong-Zakai Theorem for Dynamical Systems With Parametric Poisson White Noise Excitation
,”
Int. J. Eng. Sci.
,
40
(
10
), pp.
1165
1178
.
8.
Zhu
,
H. T.
,
2012
, “
Nonzero Mean Response of Nonlinear Oscillators Excited by Additive Poisson Impulses
,”
Nonlinear Dyn.
,
69
(
4
), pp.
2181
2191
.
9.
Guo
,
S. S.
, and
Er
,
G. K.
,
2012
, “
The Probabilistic Solution of Stochastic Oscillators With Even Nonlinearity Under Poisson Excitations
,”
Centr. Eur. J. Phys.
,
10
(
3
), pp.
702
707
.
10.
Wong
,
E.
, and
Zakai
,
M.
,
1965
, “
On the Relation Between Ordinary and Stochastic Differential Equations
,”
Int. J. Eng. Sci.
,
3
(2), pp.
1560
1564
.
11.
Di Paola
,
M.
, and
Falsone
,
G.
,
1993
, “
Stochastic Dynamics of Non-Linear Systems Driven by Non-Normal Delta-Correlated Processes
,”
ASME J. Appl. Mech.
,
60
(
1
), pp.
141
148
.
12.
Di Paola
,
M.
, and
Falsone
,
G.
,
1994
, “
Non-Linear Oscillators Under Parametric and External Poisson Pulses
,”
Nonlinear Dyn.
,
5
(
3
), pp.
337
352
.
13.
Pirrotta
,
A.
,
2007
, “
Multiplicative Cases From Additive Cases: Extension of Kolmogorov–Feller Equation to Parametric Poisson White Noise Processes
,”
Probab. Eng. Mech.
,
22
(
2
), pp.
127
135
.
14.
Yang
,
G. D.
,
Xu
,
W.
,
Feng
,
J. Q.
, and
Gu
,
X. D.
,
2015
, “
Response Analysis of Rayleigh–Van der Pol Viroimpact System With Inelastic Impact Under Two Parametric White Noise Excitations
,”
Nonlinear Dyn.
,
82
(
4
), pp.
1797
1810
.
15.
Booton
,
R. C.
,
1954
, “
Nonlinear Control Systems With Random Inputs
,”
IRE Trans. Circuit Theory
,
1
(
1
), pp.
9
18
.
16.
Caughey
,
T. K.
,
1959
, “
Response of a Nonlinear String to Random Loading
,”
ASME J. Appl. Mech.
,
26
(
3
), pp.
341
344
.http://authors.library.caltech.edu/368/1/CAUjam59a.pdf
17.
Bover
,
D. C. C.
,
1978
, “
Moment Equation Methods for Nonlinear Stochastic Systems
,”
J. Math. Anal. Appl.
,
65
(
2
), pp.
306
320
.
18.
Ibrahim
,
R. A.
,
1978
, “
Stationary Response of a Randomly Parametric Excited Nonlinear System
,”
ASME J. Appl. Mech.
,
45
(
4
), pp.
910
916
.
19.
Wu
,
W. F.
, and
Lin
,
Y. K.
,
1984
, “
Cumulant-Neglect Closure for Non-Linear Oscillator Under Random Parametric and External Excitations
,”
Int. J. Non-Linear Mech.
,
19
(
4
), pp.
349
362
.
20.
Er
,
G. K.
,
1998
, “
An Improved Closure Method for Analysis of Nonlinear Stochastic Systems
,”
Nonlinear Dyn.
,
17
(
3
), pp.
285
297
.
21.
Guo
,
S. S.
,
2014
, “
Probabilistic Solutions of Nonlinear Oscillators Excited by Correlated External and Parametric Gaussian White Noises
,”
ASME J. Vib. Acoust.
,
136
(
3
), p.
031003
.
22.
Er
,
G. K.
,
2011
, “
Methodology for the Solutions of Some Reduced Fokker–Planck Equations in High Dimensions
,”
Ann. Phys. (Berlin)
,
523
(
3
), pp.
247
258
.
23.
Zhu
,
H. T.
,
2012
, “
Probabilistic Solution of Some Multi-Degree-of-Freedom Nonlinear Systems Under External Independent Poisson White Noises
,”
J. Acoust. Soc. Am.
,
131
(
6
), pp.
4550
4557
.
24.
Shinozuka
,
M.
,
1972
, “
Monte Carlo Solution of Structural Dynamics
,”
Int. J. Numer. Methods Eng.
,
14
(5–6), pp.
855
874
.
25.
Wang
,
R.
,
Yasuda
,
K.
, and
Zhang
,
Z.
,
2000
, “
A Generalized Analysis Technique of the Stationary FPK Equation in Nonlinear System Under Gaussian White Noise Excitations
,”
Int. J. Eng. Sci.
,
38
(
12
), pp.
1315
1330
.
26.
Pirrotta
,
A.
,
2005
, “
Non-Linear Systems Under Parametric White Noise Input: Digital Simulation and Response
,”
Int. J. Non-Linear Mech.
,
40
(
8
), pp.
1088
1101
.
27.
Chaichian
,
M.
, and
Demichev
,
A.
,
2001
,
Path Integrals in Physic Volume 1: Stochastic Processes and Quantum Mechanics
,
IOP Publishing Ltd.
,
Bristol, UK
.
28.
Naess
,
A.
, and
Johnsen
,
J. M.
,
1993
, “
Response Statistics of Nonlinear, Compliant Offshore Structures by the Path Integral Solution Method
,”
Probab. Eng. Mech.
,
8
(
2
), pp.
91
106
.
29.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2012
, “
An Analytical Wiener Path Integral Technique for Non-Stationary Response Determination of Nonlinear Oscillators
,”
Probab. Eng. Mech.
,
28
, pp.
125
131
.
30.
Kougioumtzoglou
,
I. A.
,
Di Matteo
,
A.
,
Spanos
,
P. D.
,
Pirrotta
,
A.
, and
Di Paola
,
M.
,
2015
, “
An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems
,”
ASME J. App. Mech.
,
82
(
10
), p.
101005
.
31.
Shiau
,
L. C.
, and
Wu
,
T. Y.
,
1996
, “
A Finite-Element Method for Analysis of a Non-Linear System Under Stochastic Parametric and External Excitation
,”
Int. J. Non-Linear Mech.
,
31
(2), pp.
193
201
.
32.
Lin
,
Y. K.
, and
Cai
,
G. Q.
,
1995
,
Probabilistic Structural Dynamics: Advanced Theory and Applications
,
McGraw-Hill
,
New York
.
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