The characteristics of two, three, and four nonlinear vibration absorbers or nonlinear tuned mass dampers (NTMDs) attached to a structure under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. When the parameters of the absorbers are equal, it is found from the theoretical analysis that pitchfork bifurcations may occur on the part of the response curves, which are unstable in the multi-absorber systems, but are stable in a system with one NTMD. Multivalued steady-state solutions, such as three steady-state solutions for a dual-absorber system with different amplitudes, five steady-state solutions for a triple-absorber system, and seven steady-state solutions for a quadruple-absorber system, appear near bifurcation points. The NTMDs behave in that one of them vibrates at high amplitudes while the others vibrate at low amplitudes, even if the dimensions of the NTMDs are identical. Namely, “localization phenomenon” or “mode localization” occurs. After the pitchfork bifurcation, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motions, including chaotic vibrations, appear after the Hopf bifurcation when the excitation frequency decreases. Lyapunov exponents are numerically calculated to prove the occurrence of chaotic vibrations. Bifurcation sets are also calculated to investigate the influence of the system parameters on the response of the systems.

1.
Roberson
,
R. E.
, 1952, “
Synthesis of a Nonlinear Dynamic Vibration Absorber
,”
J. Franklin Inst.
0016-0032,
254
, pp.
205
220
.
2.
Pipes
,
L. A.
, 1953, “
Analysis of a Nonlinear Dynamic Vibration Absorber
,”
ASME J. Appl. Mech.
0021-8936,
20
, pp.
515
518
.
3.
Stoker
,
J. J.
, 1950,
Nonlinear Vibrations
,
Wiley
,
New York
, pp.
81
90
.
4.
Kojima
,
H.
, and
Yamakawa
,
I.
, 1980, “
Dynamic Characteristics of the Repelling Force System of the Rare-Earth Magnets (Fourth Report, Analysis of the Magnetic Dynamic Vibration Absorber With Unsymmetrical Nonlinear Restoring Force)
,”
J. Jpn. Soc. Precis. Eng.
0374-3543,
47
(
5
), pp.
54
59
.
5.
Rice
,
H. J.
, 1986, “
Combinational Instability of the Non-Linear Vibration Absorber
,”
J. Sound Vib.
0022-460X,
108
(
3
), pp.
526
532
.
6.
Natsiavas
,
S.
, 1992, “
Steady State Oscillations and Stability of Non-Linear Dynamic Vibration Absorbers
,”
J. Sound Vib.
0022-460X,
156
(
2
), pp.
227
245
.
7.
Shaw
,
J.
,
Shaw
,
S. W.
, and
Haddow
,
A. G.
, 1989, “
On the Response of the Non-Linear Vibration Absorber
,”
Int. J. Non-Linear Mech.
0020-7462,
24
(
4
), pp.
281
293
.
8.
Szemplińska-Stupnicka
,
W.
, and
Bajkowski
,
J.
, 1980, “
Multi-Harmonic Response in the Regions of Instability of Harmonic Solution in Multiple-Degree-of-Freedom Non-Linear Systems
,”
Int. J. Non-Linear Mech.
0020-7462,
15
(
1
), pp.
1
11
.
9.
Tsuda
,
Y.
,
Huang
,
M.
,
Sueoka
,
A.
, and
Tamura
,
H.
, 1997, “
Characteristics of a Nonlinear Two-Degree-of-Freedom Vibrating System
,” Preprint of the Japan Society of Mechanical Engineers, Paper No. 97-10.
10.
Zhu
,
S. J.
,
Zheng
,
Y. F.
, and
Fu
,
Y. M.
, 2004, “
Analysis of Non-Linear Dynamics of a Two-Degree-of-Freedom Vibration System With Non-Linear Damping and Non-Linear-Spring
,”
J. Sound Vib.
0022-460X,
271
(
1–2
), pp.
15
24
.
11.
Soom
,
A.
, and
Lee
,
M. -S.
, 1983, “
Optimal Design of Linear and Nonlinear Vibration Absorbers for Damped Systems
,”
ASME J. Vib., Acoust., Stress, Reliab. Des.
0739-3717,
105
(
1
), pp.
112
119
.
12.
Rice
,
H. J.
, and
McCraith
,
J. R.
, 1987, “
Practical Non-Linear Vibration Absorber Design
,”
J. Sound Vib.
0022-460X,
116
(
3
), pp.
545
559
.
13.
Jordanov
,
I. N.
, and
Cheshankov
,
B. I.
, 1988, “
Optimal Design of Linear and Non-Linear Dynamic Absorbers
,”
J. Sound Vib.
0022-460X,
123
(
1
), pp.
157
170
.
14.
Agnes
,
G. S.
, and
Inman
,
D. J.
, 2001, “
Performance of Nonlinear Vibration Absorbers for Multiple-Degree-of-Freedom Systems Using Nonlinear Normal Modes
,”
Nonlinear Dyn.
0924-090X,
25
(
1-3
), pp.
275
292
.
15.
Kojima
,
H.
, and
Saito
,
H.
, 1983, “
Forced Vibrations of a Beam With a Non-Linear Dynamic Vibration Absorber
,”
J. Sound Vib.
0022-460X,
88
(
4
), pp.
559
568
.
16.
Gendelman
,
O.
,
Manevitch
,
L. I.
,
Vakakis
,
A. F.
, and
M’Closkey
,
R.
, 2001, “
Energy Pumping in Nonlinear Mechanical Oscillators: Part I—Dynamics of the Underlying Hamiltonian Systems
,”
ASME J. Appl. Mech.
0021-8936,
68
(
1
), pp.
34
41
.
17.
Vakakis
,
A. F.
, and
Gendelman
,
O.
, 2001, “
Energy Pumping in Nonlinear Mechanical Oscillators: Part II—Resonance Capture
,”
ASME J. Appl. Mech.
0021-8936,
68
(
1
), pp.
42
48
.
18.
Abé
,
M.
, and
Fujino
,
Y.
, 1994, “
Dynamic Characterization of Multiple Tuned Mass Dampers and Some Design Formulas
,”
Earthquake Eng. Struct. Dyn.
0098-8847,
23
(
8
), pp.
813
835
.
19.
Kamiya
,
K.
,
Kamagata
,
K.
,
Matsumoto
,
S.
, and
Seto
,
K.
, 1996, “
Optimal Design Method for Multiple Dynamic Absorber
,”
Trans. Jpn. Soc. Mech. Eng., Ser. C
0387-5024,
62
(
601
), pp.
3400
3405
.
20.
Vyas
,
A.
, and
Bajaj
,
A. K.
, 2001, “
Dynamics of Autoparametric Vibration Absorbers Using Multiple Pendulums
,”
J. Sound Vib.
0022-460X,
246
(
1
), pp.
115
135
.
21.
Vyas
,
A.
, and
Bajaj
,
A. K.
, 2006, “
Global Dynamics of an Autoparametric System With Multiple Pendulums
,”
ASME J. Comput. Nonlinear Dyn.
1555-1423,
1
(
1
), pp.
35
46
.
22.
Ikeda
,
T.
, 2005, “
Nonlinear Vibrations of Elastic Structures Carrying Two Rectangular Liquid-Filled Tanks Under Horizontal Excitation
,” ASME Paper No. DETC2005-84476.
23.
Ikeda
,
T.
, 2007, “
Autoparametric Resonances in Elastic Structures Carrying Two Rectangular Tanks Partially Filled With Liquid
,”
J. Sound Vib.
0022-460X,
302
(
4–5
), pp.
657
682
.
24.
Ikeda
,
T.
,
Murakami
,
S.
, and
Ushio
,
S.
, 2009, “
Nonlinear Parametric Vibrations of Elastic Structures Containing Two Cylindrical Liquid-Filled Tanks
,”
Journal of System Design and Dynamics
,
3
(
1
), pp.
120
134
.
25.
Troger
,
H.
, and
Steindl
,
A.
, 1991,
Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Applied Scientists
,
Springer
,
New York
.
26.
Stoker
,
J. J.
, 1950,
Nonlinear Vibrations
,
Wiley
,
New York
, pp.
149
153
.
27.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
, 1995,
Applied Nonlinear Dynamics
,
Wiley
,
New York
.
28.
Pierre
,
C.
,
Tang
,
D. M.
, and
Dowell
,
E. H.
, 1987, “
Localized Vibrations of Disordered Multispan Beams
,”
AIAA J.
0001-1452,
25
(
9
), pp.
1249
1257
.
29.
Pierre
,
C.
, and
Dowell
,
E. H.
, 1987, “
Localization of Vibrations by Structural Irregularity
,”
J. Sound Vib.
0022-460X,
114
(
3
), pp.
549
564
.
30.
Vakakis
,
A. F.
, and
Cetinkaya
,
C.
, 1993, “
Mode Localization in a Class of Multidegree-of-Freedom Nonlinear Systems With Cyclic Symmetry
,”
SIAM J. Appl. Math.
0036-1399,
53
(
1
), pp.
265
282
.
31.
King
,
M. E.
, and
Vakakis
,
A. F.
, 1995, “
A Very Complicated Structure of Resonances in a Nonlinear System With Cyclic Symmetry: Nonlinear Forced Localization
,”
Nonlinear Dyn.
0924-090X,
7
(
1
), pp.
85
104
.
32.
King
,
M. E.
,
Aubrecht
,
J.
, and
Vakakis
,
A. F.
, 1995, “
Experimental Study of Steady-State Localization in Coupled Beams With Active Nonlinearities
,”
J. Nonlinear Sci.
0938-8794,
5
(
6
), pp.
485
502
.
33.
Tondl
,
A.
,
Ruijgrok
,
T.
,
Verhulst
,
F.
, and
Nabergoj
,
R.
, 2000,
Autoparametric Resonance in Mechanical Systems
,
Cambridge University Press
,
New York
.
34.
Wolf
,
A.
,
Swift
,
J. B.
,
Swinney
,
H. L.
, and
Vastano
,
J. A.
, 1985, “
Determining Lyapunov Exponents From a Time Series
,”
Physica D
0167-2789,
16
, pp.
285
317
.
35.
Doedel
,
E. J.
,
Champneys
,
A. R.
,
Fairgrieve
,
T. F.
,
Kuznetsov
,
Y. A.
,
Sandstede
,
B.
, and
Wang
,
X.
, 1997,
Continuation and Bifurcation Software for Ordinary Differential Equations (With HomCont), AUTO97
,
Concordia University
,
Canada
.
You do not currently have access to this content.