This paper is concerned with the vibration of a two degree-of-freedom (2DOF) nonlinear system subjected to multiparametric excitation forces. The vibrating motion of the system is described by the coupled differential equations having both quadratic and cubic terms. The aim of this work is to use a nonlinear absorber to control the vibration of the nonlinear system near the simultaneous subharmonic and internal resonances, where the vibrations are severe. Multiple scale perturbation technique (MSPT) is applied to obtain the averaged equations up to the second-order approximation. The steady-state response and their stability are studied numerically for the nonlinear system at the simultaneous subharmonic and internal resonances. Some recommendations regarding to the different system parameters are given following studying the effects of various parameters. Comparison with the available published work is made.

References

References
1.
Mead
,
D. J.
,
1988
,
Passive Vibration Control
,
Wiley
,
Chichester, UK
.
2.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
3.
Eissa
,
M.
,
1999
, “
Vibration Control of Non-Linear Mechanical System via a Neutralizer
,” Faculty of Electronic Engineering, Menouf, Egypt, Electronic Bulletin No. 18.
4.
Tondl
,
A.
, and
Nabergoj
,
R.
,
2000
, “
Dynamic Absorbers for an Externally Excited Pendulum
,”
J. Sound Vib.
,
234
(
4
), pp.
611
624
.10.1006/jsvi.1999.2892
5.
Ji
,
J. C.
, and
Zhang
,
N.
,
2010
, “
Suppression of the Primary Resonance Vibrations of a Forced Nonlinear System Using a Dynamic Vibration Absorber
,”
J. Sound Vib.
,
329
(
11
), pp.
2044
2056
.10.1016/j.jsv.2009.12.020
6.
Ji
,
J. C.
, and
Zhang
,
N.
,
2011
, “
Suppression of Super-Harmonic Resonance Response Using a Linear Vibration Absorber
,”
Mech. Res. Commun.
,
38
(
6
), pp.
411
416
.10.1016/j.mechrescom.2011.05.014
7.
Amer
,
Y. A.
, and
El-Sayed
,
A. T.
,
2008
, “
Vibration Suppression of Non-Linear System via Non-Linear Absorber
,”
Commun. Nonlinear Sci. Numer. Simul.
,
13
(
9
), pp.
1948
1963
.10.1016/j.cnsns.2007.04.018
8.
Kamel
,
M.
,
Eissa
,
M.
, and
El-Sayed
,
A. T.
,
2009
, “
Vibration Reduction of a Non-Linear Spring Pendulum Under Multi-Parametric Excitations via a Longitudinal Absorber
,”
Phys. Scr.
,
80
(
12
), p.
025005
.10.1088/0031-8949/80/02/025005
9.
Eissa
,
M.
,
Kamel
,
M.
, and
El-Sayed
,
A. T.
,
2010
, “
Vibration Reduction of Multi-Parametric Excited Spring Pendulum via a Transversally Tuned Absorber
,”
Nonlinear Dyn.
,
61
(
1–2
), pp.
109
121
.10.1007/s11071-009-9635-4
10.
El-Sayed
,
A. T.
,
Kamel
,
M.
, and
Eissa
,
M.
,
2010
, “
Vibration Reduction of a Pitch-Roll Ship Model With Longitudinal and Transverse Absorbers Under Multi Excitations
,”
Math. Comput. Modell.
,
52
(
9–10
), pp.
1877
1898
.10.1016/j.mcm.2010.07.027
11.
Eissa
,
M.
,
Kamel
,
M.
, and
El-Sayed
,
A. T.
,
2011
, “
Vibration Reduction of a Nonlinear Spring Pendulum Under Multi External and Parametric Excitations via a Longitudinal Absorber
,”
Meccanica
,
46
(
2
), pp.
325
340
.10.1007/s11012-010-9311-2
12.
Eissa
,
M.
,
Kamel
,
M.
, and
El-Sayed
,
A. T.
,
2012
, “
Vibration Suppression of a 4-DOF Nonlinear Spring Pendulum via a Longitudinal and a Transverse Absorbers
,”
ASME J. Appl. Mech.
,
79
(
1
), p.
011007
.10.1115/1.4004551
13.
Eissa
,
M.
,
Amer
,
Y. A.
, and
Bauomy
,
H. S.
,
2007
, “
Active Control of an Aircraft Tail Subject to Harmonic Excitation
,”
J. Acta Mech. Sin.
,
23
(
4
), pp.
451
462
.10.1007/s10409-007-0077-2
14.
Amer
,
Y. A.
, and
Bauomy
,
H. S.
,
2009
, “
Vibration Reduction in a 2DOF Twin Tail System to Parametric Excitations
,”
Commun. Nonlinear Sci. Numer. Simul.
,
14
(
1
), pp.
560
573
.10.1016/j.cnsns.2007.10.005
15.
Amer
,
Y. A.
,
Bauomy
,
H. S.
, and
Sayed
,
M.
,
2009
, “
Vibration Suppression in a Twin-Tail System to Parametric and External Excitations
,”
Comput. Math. Appl.
,
58
(
10
), pp.
1947
1964
.10.1016/j.camwa.2009.07.090
16.
Kamel
,
M.
, and
Bauomy
,
H. S.
,
2009
, “
Nonlinear Oscillation of a Rotor-AMB System With Time Varying Stiffness and Multi-External Excitations
,”
ASME J. Vib. Acoust.
,
131
(
3
), pp.
1
11
.10.1115/1.3085884
17.
Kamel
,
M.
, and
Bauomy
,
H. S.
,
2010
, “
Nonlinear Behavior of a Rotor-AMB System Under Multi-Parametric Excitations
,”
Meccanica
,
45
(
1
), pp.
7
22
.10.1007/s11012-009-9213-3
18.
Eissa
,
M.
,
Kamel
,
M.
, and
Bauomy
,
H. S.
,
2011
, “
Nonlinear Behavior of Tuned Rotor-AMB System With Time Varying Stiffness
,”
J. Bifurcation Chaos
,
21
(
1
), pp.
195
207
.10.1142/S0218127411028362
19.
Eissa
,
M.
,
Kamel
,
M.
, and
Bauomy
,
H. S.
,
2011
, “
Dynamics of Tuned an AMB–Rotor With Time Varying Stiffness and Mixed Excitations
,”
Meccanica
,
47
(
3
), pp.
585
601
.10.1007/s11012-011-9469-2
20.
El-Ganaini
,
W.
, and
El-Gohary
,
H. A.
,
2014
, “
Application of Time-Delay Absorber to Suppress Vibration of a Dynamical System to Tuned Excitation
,”
ASME J. Vib. Acoust.
,
136
(
4
), pp.
1
10
.10.1115/1.4027629
21.
Nayfeh
,
A. H.
,
1973
,
Perturbation Methods
,
Wiley
,
New York
.
22.
Lee
,
W. K.
, and
Park
,
H. D.
,
1997
, “
Chaotic Dynamics of a Harmonically Excited Spring Pendulum System With Internal Resonance
,”
Nonlinear Dyn.
,
14
(
3
), pp.
211
229
.10.1023/A:1008256920441
You do not currently have access to this content.