This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

References

References
1.
Zuo
,
L.
, and
Nayfeh
,
S. A.
,
2006
, “
The Two-Degree-of-Freedom Tuned-Mass Damper for Suppression of Single-Mode Vibration Under Random and Harmonic Excitation
,”
ASME J. Vib. Acoust.
,
128
(
1
), pp.
56
65
.10.1115/1.2128639
2.
Dimentberg
,
M. F.
, Jr.
,
1976
, “
Response of a Non-Linearly Damped Oscillator to Combined Periodic Parametric and Random External Excitation
,”
Int. J. Non-Linear Mech.
,
11
(
1
), pp.
83
87
.10.1016/0020-7462(76)90040-8
3.
Iyengar
,
R. N.
,
1986
, “
A Nonlinear System Under Combined Periodic and Random Excitation
,”
J. Stat. Phys.
,
44
(
5–6
), pp.
907
920
.10.1007/BF01011913
4.
Manohar
,
C. S.
, and
Iyengar
,
R. N.
,
1991
, “
Entrainment in Van der Pol's Oscillator in the Presence of Noise
,”
Int. J. Non-Linear Mech.
,
26
(
5
), pp.
679
686
.10.1016/0020-7462(91)90019-P
5.
Nayfeh
,
A. H.
, and
Serhan
,
S. J.
,
1990
, “
Response Statistics of Non-Linear Systems to Combined Deterministic and Random Excitations
,”
Int. J. Non-Linear Mech.
,
25
(
5
), pp.
493
509
.10.1016/0020-7462(90)90014-Z
6.
Serhan
,
S. J.
, and
Nayfeh
,
A. H.
,
1991
, “
Nonlinear Random Coupled Motions of Structural Elements With Quadratic Nonlinearities
,”
Nonlinear Dyn.
,
2
(
4
), pp.
305
316
.10.1007/BF00045299
7.
Cai
,
G. Q.
, and
Lin
,
Y. K.
,
1994
, “
Nonlinearly Damped Systems Under Simultaneous Broad-Band and Harmonic Excitations
,”
Nonlinear Dyn.
,
6
(
2
), pp.
163
177
.10.1007/BF00044983
8.
Rong
,
H.
,
Xu
,
W.
, and
Fang
,
T.
,
1998
, “
Principal Response of Duffing Oscillator to Combined Deterministic and Narrow-Band Random Parametric Excitation
,”
J. Sound Vib.
,
210
(
4
), pp.
483
515
.10.1006/jsvi.1997.1325
9.
Huang
,
Z. L.
,
Zhu
,
W. Q.
, and
Suzuki
,
Y.
,
2000
, “
Stochastic Averaging of Strongly Non-Linear Oscillators Under Combined Harmonic and White-Noise Excitations
,”
J. Sound Vib.
,
238
(
2
), pp.
233
256
.10.1006/jsvi.2000.3083
10.
Haiwu
,
R.
,
Wei
,
X.
,
Guang
,
M.
, and
Tong
,
F.
,
2001
, “
Response of a Duffing Oscillator to Combined Deterministic Harmonic and Random Excitation
,”
J. Sound Vib.
,
242
(
2
), pp.
362
368
.10.1006/jsvi.2000.3329
11.
Zhu
,
W. Q.
, and
Wu
,
Y. J.
,
2003
, “
First-Passage Time of Duffing Oscillator Under Combined Harmonic and White-Noise Excitations
,”
Nonlinear Dyn.
,
32
(
3
), pp.
291
305
.10.1023/A:1024414020813
12.
Haiwu
,
R.
,
Guang
,
M.
,
Xiangdong
,
W.
,
Wei
,
X.
, and
Tong
,
F.
,
2004
, “
Response Statistic of Strongly Non-Linear Oscillator to Combined Deterministic and Random Excitation
,”
Int. J. Non-Linear Mech.
,
39
(
6
), pp.
871
878
.10.1016/S0020-7462(03)00070-2
13.
Xu
,
W.
,
He
,
Q.
,
Fang
,
T.
, and
Rong
,
H.
,
2004
, “
Stochastic Bifurcation in Duffing System Subject to Harmonic Excitation and in Presence of Random Noise
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1473
1479
.10.1016/j.ijnonlinmec.2004.02.009
14.
Yu
,
J. S.
, and
Lin
,
Y. K.
,
2004
, “
Numerical Path Integration of a Non-Homogeneous Markov Process
,”
Int. J. Non-Linear Mech.
,
39
(
9
), pp.
1493
1500
.10.1016/j.ijnonlinmec.2004.02.011
15.
Benedettini
,
F.
,
Zulli
,
D.
, and
Vasta
,
M.
,
2006
, “
Nonlinear Response of SDOF Systems Under Combined Deterministic and Random Excitations
,”
Nonlinear Dyn.
,
46
(
4
), pp.
375
385
.10.1007/s11071-006-9029-9
16.
Xie
,
W. X.
,
Xu
,
W.
, and
Cai
,
L.
,
2006
, “
Study of the Duffing-Rayleigh Oscillator Subject to Harmonic and Stochastic Excitations by Path Integration
,”
Appl. Math. Comput.
,
172
(
2
), pp.
1212
1224
.10.1016/j.amc.2005.03.018
17.
Wu
,
Y. J.
, and
Zhu
,
W. Q.
,
2008
, “
Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations
,”
ASME J. Vib. Acoust.
,
130
(
5
), p.
051004
.10.1115/1.2948382
18.
Chen
,
L. C.
, and
Zhu
,
W. Q.
,
2009
, “
Stochastic Averaging of Strongly Nonlinear Oscillators With Small Fractional Derivative Damping Under Combined Harmonic and White Noise Excitations
,”
Nonlinear Dyn.
,
56
(
3
), pp.
231
241
.10.1007/s11071-008-9395-6
19.
Rong
,
H.
,
Wang
,
X.
,
Xu
,
W.
, and
Fang
,
T.
,
2010
, “
Resonant Response of a Non-Linear Vibro-Impact System to Combined Deterministic Harmonic and Random Excitations
,”
Int. J. Non-Linear Mech.
,
45
(
4
), pp.
474
481
.10.1016/j.ijnonlinmec.2010.01.005
20.
Narayanan
,
S.
, and
Kumar
,
P.
,
2012
, “
Numerical Solutions of Fokker-Planck Equation of Nonlinear Systems Subjected to Random and Harmonic Excitations
,”
Probab. Eng. Mech.
,
27
(
1
), pp.
35
46
.10.1016/j.probengmech.2011.05.006
21.
Anh
,
N. D.
, and
Hieu
,
N. N.
,
2012
, “
The Duffing Oscillator Under Combined Periodic and Random Excitations
,”
Probab. Eng. Mech.
,
30
, pp.
27
36
.10.1016/j.probengmech.2012.02.004
22.
Anh
,
N. D.
,
Zakovorotny
,
V. L.
, and
Hao
,
D. N.
,
2014
, “
Response Analysis of Van der Pol Oscillator Subjected to Harmonic and Random Excitations
,”
Probab. Eng. Mech.
,
37
, pp.
51
59
.10.1016/j.probengmech.2014.05.001
23.
Cheung
,
Y. K.
, and
Iu
,
V. P.
,
1988
, “
An Implicit Implementation of Harmonic Balance Method for Non-Linear Dynamic Systems
,”
Eng. Comput.
,
5
(
2
), pp.
134
140
.10.1108/eb023731
24.
Iu
, V
. P.
, and
Cheung
,
Y. K.
,
1986
, “
Non-Linear Vibration Analysis of Multilayer Sandwich Plates by Incremental Finite Elements: 2. Solution Techniques and Examples
,”
Eng. Comput.
,
3
(
1
), pp.
43
52
.10.1108/eb023640
25.
Clough
,
R. W.
, and
Penzien
,
J.
,
2003
,
Dynamics of Structures
,
3rd ed.
,
Computers & Structures
,
Berkeley, CA
.
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