In this paper, a new strategy based on generalized cell mapping (GCM) method will be introduced to investigate the stochastic response of a class of impact systems. Significant difference of the proposed procedure lies in the choice of a novel impact-to-impact mapping, which is built to calculate the one-step transition probability matrix, and then, the probability density functions (PDFs) of the stochastic response can be obtained. The present strategy retains the characteristics of the impact systems, and is applicable to almost all types of impact systems indiscriminately. Further discussion proves that our strategy is reliable for different white noise excitations. Numerical simulations verify the efficiency and accuracy of the suggested strategy.

References

References
1.
di Bernardo
,
M.
,
Budd
,
C. J.
,
Champneys
,
A. R.
, and
Kpwalczyk
,
P.
,
2008
,
Piecewise-Smooth Dynamical Systems Theory and Applications
,
Springer
,
Berlin
.
2.
Shaw
,
S. W.
,
1989
, “
The Transition to Chaos in a Simple Mechanical System
,”
Non-Linear Mech.
,
24
(
1
), pp.
41
56
.
3.
Nordmark
,
A. B.
,
1991
, “
Nonperiodic Motion Caused by Grazing-Incidence in an Impact Oscillator
,”
J. Sound Vib.
,
145
(
2
), pp.
279
297
.
4.
Wai
,
C.
,
Edward
,
O.
,
Helena
,
N.
, and
Celso
,
E. G.
,
1994
, “
Grazing Bifurcations in Impact Oscillators
,”
Phys. Rev. E
,
50
(
6
), pp.
4427
4444
.
5.
Brogliatoa
,
B.
,
1999
,
Non-Smooth Mechanics: Model, Dynamics and Control
,
Springer
,
Berlin
.
6.
Edward
,
O.
,
2002
,
Chaos in Dynamical Systems
,
Cambridge University Press
,
Cambridge, UK
.
7.
Luo
,
G. W.
,
2004
, “
Period-Doubling Bifurcations and Routes to Chaos of the Vibratory Systems Contacting Stops
,”
Phys. Lett. A
,
323
(
3–4
), pp.
210
217
.
8.
Luo
,
G. W.
,
Xie
,
J. H.
, and
Zhu
,
X. F.
,
2008
, “
Periodic Motions and Bifurcations of a Vibro-Impact System
,”
Chaos, Solitons Fractals
,
36
(
5
), pp.
1340
1347
.
9.
Yue
,
Y.
,
Miao
,
P. C.
,
Xie
,
J. H.
, and
Celso
,
G.
,
2016
, “
Symmetry Restoring Bifurcations and Quasiperiodic Chaos Induced by a New Intermittency in a Vibro-Impact System
,”
Chaos
,
26
(
11
), p.
113121
.
10.
Hu
,
H. Y.
,
1995
, “
Numerical Scheme of Location the Periodic Response of Non-Smooth Non-Autonomous Systems of High Dimension
,”
Comput. Methods Appl. Mech. Eng.
,
123
(
1–4
), pp.
53
62
.
11.
Feng
,
J. Q.
, and
Xu
,
W.
,
2008
, “
Stochastic Responses of Vibro-Impact Duffing Oscillator Excited by Additive Gaussian Noise
,”
J. Sound Vib.
,
309
(
3–5
), pp.
730
738
.
12.
Dimentberg
,
M. F.
, and
Iourtchenko
,
D. V.
,
1999
, “
Towards Incorporating Impact Losses Into Random Vibration Analyses: A Model Problem
,”
Probab. Eng. Mech.
,
14
(
4
), pp.
323
328
.
13.
Dimentberg
,
M. F.
,
Gaidai
,
O.
, and
Naess
,
A.
,
2009
, “
Random Vibrations With Strongly Inelastic Impacts: Response PDF by the Path Integration Method
,”
Int. J. Non-Linear Mech.
,
44
(
7
), pp.
791
796
.
14.
Paola
,
M. D.
, and
Bucher
,
C.
,
2016
, “
Ideal and Physical Barrier Problems for Non-Linear Systems Driven by Normal and Poissonian White Noise Via Path Integral Method
,”
Int. J. Non-Linear Mech.
,
81
, pp.
274
282
.
15.
Iourtchenko
,
D.
,
Mo
,
E.
, and
Naess
,
A.
,
2008
, “
Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method
,”
ASME J. Appl. Mech.
,
75
(
6
), p.
061016
.
16.
Hsu
,
C. S.
,
1987
,
Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear System
,
Springer-Verlag
,
New York
.
17.
Hong
,
L.
, and
Xu
,
J. X.
,
1999
, “
Crises and Chaotic Transients Studied by the Generalized Cell Mapping Digraph Method
,”
Phys. Lett. A
,
262
(
4–5
), pp.
361
375
.
18.
Gaull
,
A.
, and
Kreuzer
,
E.
,
2007
, “
Cell Mapping Applied to Random Dynamical Systems
,”
IUTAM Symposium on Dynamics and Control of Nonlinear Systems With Uncertainty
, Nanjing, China, Sept. 18–22, pp.
65
76
.
19.
Sun
,
J. Q.
, and
Hsu
,
C. S.
,
1990
, “
The Generalized Cell Mapping Method in Nonlinear Random Vibration Based Upon Short-Time Gaussian Approximation
,”
ASME J. Appl. Mech.
,
57
(
4
), pp.
1018
1025
.
20.
Yue
,
X. L.
,
Xu
,
W.
,
Jia
,
W. T.
, and
Wang
,
L.
,
2013
, “
Stochastic Response of a ϕ6 Oscillator Subjected to Combined Harmonic and Poisson White Noise Excitations
,”
Physica A
,
392
(
14
), pp.
2988
2998
.
21.
Li
,
Z. G.
,
Jun
,
J.
, and
Hong
,
L.
,
2017
, “
Noise-Induced Transition in a Piecewise Smooth System by Generalized Cell Mapping Method With Evolving Probabilistic Vector
,”
Nonlinear Dyn.
,
88
(
2
), pp.
1473
1485
.
22.
Gan
,
C. B.
, and
Lei
,
H.
,
2011
, “
Stochastic Dynamical Analysis of a Kind of Vibro-Impact System Under Multiple Harmonic and Random Excitations
,”
J. Sound Vib.
,
330
(
10
), pp.
2174
2184
.
23.
Zhu
,
W. Q.
,
1998
,
Stochastic Vibration
,
Science and Technology Press
,
Beijing, China
.
24.
Li
,
J.
, and
Zhang
,
S. J.
,
2007
, “
Cell-Mapping Computation Method for Non-Smooth Dynamical Systems
,”
ACTA Mech. Solida Sin.
,
28
(
1
), pp.
93
96
.
25.
Di Paola
,
M.
, and
Santoro
,
R.
,
2008
, “
Path Integral Solution for Non-Linear System Enforced by Poisson White Noise
,”
Probab. Eng. Mech.
,
23
(
2–3
), pp.
164
169
.
26.
Pirrotta
,
A.
, and
Santoro
,
R.
,
2011
, “
Probabilistic Response of Nonlinear Systems Under Combined Normal and Poisson White Noise Via Path Integral Method
,”
Probab. Eng. Mech.
,
26
(
1
), pp.
26
32
.
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