Two principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer–Rosenberg and Shaw–Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described.

References

References
1.
Lyapunov
,
A. M.
, 1947,
The General Problem of the Stability of Motion
,
Princeton University
,
Princeton, NJ
.
2.
Kauderer
,
H.
, 1958,
Nichtlineare Mechanik
,
Springer-Verlag
,
Berlin
.
3.
Seifert
,
H.
, 1948, “
Periodische bewegungen mechaischen systeme
,”
Math. Z.
,
51
, pp.
197
216
.
4.
Rosenberg
,
R. M.
, and
Atkinson
,
C. P.
, 1959, “
On the Natural Modes and Their Stability in Nonlinear Two-degree-of-freedom Systems
,”
ASME J. Appl. Mech.
,
26
, pp.
377
385
.
5.
Rosenberg
,
R.
, 1960, “
Normal Modes of Nonlinear Dual Mode Systems
,”
ASME J. Appl. Mech.
,
27
, pp.
263
268
.
6.
Rosenberg
,
R.
, 1961, “
On Normal Vibration of a General Class of Nonlinear Dual Mode Systems
,”
ASME J. Appl. Mech.
28
, pp.
275
283
.
7.
Rosenberg,
R.
, 1962, “
The Normal Modes of Nonlinear n-Degree-of-Freedom Systems
,”
ASME J. Appl. Mech.
,
29
, pp.
7
14
.
8.
Rosenberg
,
R.
, 1964, “
On Normal Mode Vibration
,”
Proc. Cambridge. Philos. Soc.
60
, pp.
595
611
.
9.
Rosenberg
,
R.
, 1966, “
Nonlinear Vibrations of Systems With Many Degrees of Freedom
,”
Adv. Appl. Mech.
,
9
, pp.
156
243
.
10.
Rosenberg
,
R.
, and
Kuo
,
J.
, 1964, “
Nonsimilar Normal Mode Vibrations of Nonlinear Systems Having Two Degrees of Freedom
,”
ASME J. Appl. Mech.
,
31
, pp.
283
290
.
11.
Rand
,
R.
, 1971, “
A Higher Order Approximation for Nonlinear Normal Modes in Two Degrees of Freedom Systems
,”
Int. J. Non-Linear Mech.
,
6
, pp.
545
547
.
12.
Manevich
,
L.
, and
Mikhlin
,
Y.
, 1972, “
Periodic Solutions Close to Rectilinear Normal Vibration Modes
,”
Prikl. Mat. Mekh
.,
36
, pp.
1051
1058
.
13.
Mikhlin
,
Y.
, 1996, “
Normal Vibrations of a General Class of Conservative Oscillators
,”
Nonlinear. Dyn.
,
11
, pp.
1
16
.
14.
Manevich
,
L.
,
Mikhlin
,
Y.
, and
Pilipchuk
,
V.
, 1989,
The Method of Normal Oscillation for Essentially Nonlinear Systems
,
Nauka
,
Moscow (in Russian)
.
15.
Vakakis
,
A.
,
Manevitch
,
L.
,
Mikhlin
,
Y.
,
Pilipchuk
,
V.
, and
Zevin
,
A.
, 1996,
Normal Modes and Localization in Nonlinear Systems
,
Wiley
,
New York
.
16.
Vakakis
,
A.
, and,
R. H.
Rand
1992, “
Normal Modes and Global Dynamics of a Two-Degree-of Freedom Non-Linear System. I. Low Energies
,”
Int. J. Non-Linear Mech.
,
27
, pp.
861
888
.
17.
Mikhlin
,
Y.
, 1995, “
Matching of Local Expansions in the Theory of Nonlinear Vibrations
,”
J. Sound Vib.
,
182
, pp.
577
588
.
18.
Shaw
,
S. W.
, and
Pierre
,
C.
, 1994, “
Normal Modes of Vibration for Non-Linear Continuous Systems
,”
J. Sound Vib
.,
169
(
3
), pp.
319
347
.
19.
Pesheck
,
E.
,
Pierre
,
C.
, and
Shaw
,
S. W.
, 2002, “
A New Galerkin-Based Approach for Accurate Non-Linear Normal Modes Through Invariant Manifolds
,”
J. Sound Vib.
,
249
(
5
), pp.
971
993
.
20.
Boivin
,
N.
,
Pierre
,
C.
, and
Shaw
,
S. W.
, 1995, “
Non-Linear Modal Analysis of Structural Systems Featuring Internal Resonances
,”
J. Sound Vib
.,
182
(
2
),pp.
336
341
.
21.
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S. W.
, 2005, “
The Construction of Non-Linear Normal Modes for Systems With Internal Resonances
,”
Int. J. Non-Linear Mech.
,
40
, pp.
729
746
.
22.
Malkin
,
I.
, 1956,
Some Problems of the Theory of Nonlinear Vibrations
,
Geotechteorizdat
,
Moscow (in Russian)
.
23.
Kinney
,
W.
, and
Rosenberg
,
R.
, 1966, “
On the Steady State Vibrations of Nonlinear Systems With Many Degrees of Freedom
,”
ASME J. Appl. Mech
.,
33
, pp.
406
412
.
24.
Mikhlin
,
Y.
, 1974, “
Resonance Modes of Near-Conservative Nonlinear Systems
,”
Prikl. Mat. Mekh.
,
38
, pp.
425
429
.
25.
Mikhlin
,
Y.
, and
Morgunov
,
B.
, 2001, “
Normal Vibrations in Near-Conservative Self-Exited and Viscoelastic Nonlinear Systems
,”
Nonlinear Dyn
.,
25
, pp.
33
48
.
26.
Avramov
,
K. V.
, 2009, “
Nonlinear Modes of Parametric Vibrations and Their Applications to Beams Dynamics
,”
J. Sound Vib
.,
322
(
3
), pp.
476
489
.
27.
Mishra
,
A. K.
, and
Singh
,
M. C.
, 1974, “
The Normal Modes of Nonlinear Symmetric Systems by Group Representation Theory
,”
Int. J. Nonlinear Mech
.,
9
, pp.
463
480
.
28.
Chechin
,
G. M.
,
Sakhnenko
,
V. P.
,
Stokes
,
H. T.
,
Smith
,
A. D.
, and
Hatch
,
D. M.
, 2000, “
Non-Linear Normal Modes for Systems With Discrete Symmetry
,”
Int. J. Nonlinear Mech
.,
35
, pp.
497
513
.
29.
Chechin
,
G. M.
, and
Dzhelauhova
,
G. S.
, 2009, “
Discrete Breathers and Nonlinear Normal Modes in Monoatomic Chains
,”
J. Sound Vib
.,
322
(
3
), pp.
490
512
.
30.
Vedenova
,
E.
,
Manevich
,
L.
, and
Pilipchuk
,
V.
, 1985, “
Normal Oscillations of a String With Concentrated Masses on Non-Linearly Elastic Support
,”
Prikl.
Matem
Mekh
.,
49
, pp.
203
211
.
31.
Zuo
,
L.
, and
Curnier
,
A.
, 1994, “
Non-Linear Real and Complex Modes of Conewise Linear Systems
,”
J. Sound Vib
.,
174
(
3
), pp.
289
313
.
32.
Pilipchuk
,
V. N.
, 2001, “
Impact Modes in Discrete Vibrating Systems With Rigid Barriers
,”
Int. J. Nonlinear Mech
.,
36
, pp.
999
1012
.
33.
Avramov
,
K. V.
,
Mikhlin
,
Y.
, and
Kurilov
,
E.
, 2007, “
Asymptotic Analysis of Nonlinear Dynamics of Simply Supported Cylindrical Shells
,”
Nonlinear Dyn
.,
47
, pp.
331
352
.
34.
Belizzi
,
S.
, and
Bouc
,
R.
, 2005, “
A New Formulation for the Existence and Calculation of Nonlinear Normal Modes
,”
J. Sound Vib
.,
287
(
3
), pp.
545
569
.
35.
King
,
M. E.
, and
Vakakis
,
A. F.
, 1993, “
An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems
,”
ASME J. Vibr. Acoust
.
116
(
3
), pp.
332
340
.
36.
Nayfeh
,
A.
, and
Nayfeh
,
S.
, 1994, “
On Nonlinear Modes of Continuous Systems
,”
ASME J. Vibr. Acoust.
116
, pp.
129
136
.
37.
Carr
,
J.
, 1981,
Applications of Centre Manifold Theory
,
Springer-Verlag
,
New York
.
38.
Avramov
,
K. V.
, “
Many-Dimensional Models of Traveling Waves and Nonlinear Modes in Cylindrical Shell
,” Int. Appl. Mech. (in press).
39.
Breslavsky
,
I. D.
,
Strel’nikova
,
E. A.
, and
Avramov
,
K. V.
, 2011, “
Dynamics of Shallow Shells With Geometrical Nonlinearity Interacting With Fluid
,”
Comp. and Struc.
,
89
, pp.
496
506
.
40.
Vakakis
,
A. F.
,
Manevitch
,
L. I.
,
Gendelman
,
O.
, and
Bergman
,
L.
, 2003, “
Dynamics of Linear Discrete Systems Connected to Local, Essentially Non-Linear Attachments
,”
J. Sound Vibr.
,
264
, pp.
559
577
.
41.
Gendelman
,
O. V.
, 2004, “
Bifurcations of Nonlinear Normal Modes of Linear Oscillator With Strongly Nonlinear Damped Attachment
,”
Nonlinear Dyn.
,
37
(
2
), pp.
115
128
.
42.
Avramov
,
K. V.
, and
Mikhlin
,
Y.
, 2004, “
Snap-Through Truss as a Vibration Absorber
,”
J. Vib. Control
10
, pp.
291
308
.
43.
Avramov
,
K. V.
, and
Mikhlin
,
Y.
, 2004, “
Forced Oscillations of a System Containing a Snap-Through Truss, Close to Its Equilibrium Position
,”
Nonlinear Dyn.
,
35
, pp.
361
379
.
44.
Avramov
,
K. V.
, and
Mikhlin
,
Y.
, 2006, “
Snap-Through Truss as an Absorber of Forced Oscillations
,”
J. Sound Vib.
,
29
, pp.
705
722
.
45.
Mikhlin
,
Y.
, and
Reshetnikova
,
S. N.
, 2005, “
Dynamical Interaction of an Elastic System and Essentially Nonlinear Absorber
,”
J. Sound Vibr.
,
283
, pp.
91
120
.
46.
Breslavsky
,
I.
,
Avramov
,
K. V.
,
Mikhlin
,
Y.
, and
Kochurov
,
R.
, 2008, “
Nonlinear Modes of Snap-Through Motions of a Shallow Arch
,”
J. Sound Vib.
,
311
, pp.
297
313
.
47.
Breslavsky
,
I.
, and
Avramov
,
K. V.
Nonlinear Modes of Cylindrical Panels With Complex Boundaries. R-Function Method
,” Meccanica (in press).
48.
Kochurov
,
R.
, and
Avramov
,
K. V.
, 2010, “
Nonlinear Modes and Traveling Waves of Parametrically Excited Cylindrical Shells
,”
J. Sound Vibr.
,
329
, pp.
2193
2204
.
49.
Legrand
,
M.
,
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S. W.
, 2004, “
Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method
,”
Int. J. Rotating Mach.
,
10
(
4
), pp.
319
335
.
50.
Avramov
,
K. V.
, 2010, “
Nonlinear Modes of Self-Sustained Vibrations of One Disk Rotor in Two Journal Bearings
,”
Strength Mater
.
4
, pp.
130
144
.
51.
Mikhlin
,
Y.
, and
Mitrokhin
,
S.
2008, “
Nonlinear Vibration Modes of the Double Tracked Road Vehicle
,”
J. Theor. Appl. Mech.
,
46
(
3
), pp.
581
596
.
52.
Manevitch
,
L. I.
, 2001, “
The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables
,”
Nonlinear Dyn.
,
25
, pp.
95
109
.
53.
Vakakis
,
A. F.
, 2001, “
Inducing Passive Nonlinear Energy Sinks in Vibrating Systems
,”
ASME J. Vib. Acoust.
123
(
3
), pp.
324
32
.
54.
Gendelman
,
O.
, 2001, “
Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators
,”
Nonlinear Dyn.
,
25
, pp.
237
253
.
55.
Gendelman
,
O. V.
,
Vakakis
,
A. F.
,
Manevitch
,
L. I.
, and
McCloskey
,
R.
, 2001, “
Energy Pumping in Nonlinear Mechanical Oscillators I: Dynamics of the Underlying Hamiltonian System
,”
ASME J. Appl. Mech.
,
68
, pp.
34
41
.
56.
Gendelman
,
O. V.
,
Gorlov
,
D. V.
,
Manevitch
,
L. I.
, and
Musienko
,
A. I.
, 2005, “
Dynamics of Coupled Linear and Essentially Nonlinear Oscillators With Substantially Different Masses
,”
J. Sound Vibr.
,
286
, pp.
1
19
.
57.
Vakakis
,
A. F.
,
Gendelman
,
O. V.
,
Bergman
,
L. A.
,
McFarland
,
D. M.
,
Kerschen
,
G.
, and
Lee
,
Y. S.
, 2008,
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems
(
Solid Mechanics and Its Applications
, Vol.
156
),
Springer-Verlag
,
Berlin
.
58.
Pilipchuk
,
V. N.
, 2009, “
Transition From Normal to Local Modes in an Elastic Beam Supported by Nonlinear Springs
,”
J. Sound Vib.
,
322
, pp.
554
563
.
59.
Pilipchuk
,
V. N.
2010,
Nonlinear Dynamics: Between Linear and Impact Limits
,
Springer–Verlag
,
Berlin
.
60.
Avramov
,
K. V.
, and
Mikhlin
,
Y.
, 2010,
Nonlinear Dynamics of Elastic Systems, Vol. 1: Models, Methods, and Approaches
(
Regular and Chaotic Dynamics
),
Scientific Centre
,
Moscow
(in Russian).
61.
Vakakis
,
A. F.
, 1997, “
Non-Linear Normal Modes (NNMs) and Their Applications in Vibration Theory: An Overview
,”
Mech. Syst. Signal Process
.
11
(
1
), pp.
3
22
.
62.
Kerschen
,
G.
,
Peeters
,
M.
,
Golinval
,
J. C.
, and
Vakakis
,
A. F.
, 2009, “
Nonlinear Normal Modes, Part I: A Useful Framework for the Structural Dynamicist
,”
Mech. Syst.
Signal Process.
23
, pp.
170
194
.
63.
Lanczos
,
C.
, 1962,
The Variational Principles of Mechanics
,
University of Toronto
,
Toronto
.
64.
Lusternik
,
L. A.
, and
Shnirel’man
,
L. G.
, 1930,
Topological Methods in Nonlinear Sciences
,
GNTI
,
Moscow
(in Russian)
65.
van Groesen
,
E. W. C.
, 1983, “
On Normal Modes in Classical Hamiltonian Systems
,”
Int. J. Nonlinear Mech.
,
18
(
1
), pp.
55
70
.
66.
Atkinson
,
C. P.
,
Bhatt
,
S. J.
, and
Pacitti
,
T.
, 1963, “
The Stability of the Normal Modes of Nonlinear Systems With Polynomial Restoring Forces of High Degree
,”
ASME J. Appl. Mech.
,
30
, pp.
163
198
.
67.
Rosenberg
,
R. M.
, and
Hsu
,
C. S.
, 1963, “
On the Geometrization of Normal Vibrations of Nonlinear Systems Having Many Degrees of Freedom
,”
Proceedings of the International Symposium on Nonlinear Vibrations IUTAM
,
Kiev, Ukraine
, Vol.
1
, pp.
380
416
.
68.
Blaquiere
,
A.
1966,
Nonlinear System Analysis
,
Academic Press
,
New York
.
69.
King
,
M. E.
, and
Vakakis
,
A. F.
, 1996, “
An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes
,”
ASME J. Appl. Mech.
63
, pp.
810
819
.
70.
Yang
,
T. L.
, and
Rosenberg
,
R. M.
, 1967, “
On the Vibrations of a Particle in the Plane
,”
Int. J. Nonlinear Mech.
,
2
(
1
), pp.
1
25
.
71.
Rand
,
R. H.
, 1974, “
A Direct Method for Non-Linear Normal Modes
,”
Int. J. Nonlinear Mech.
,
9
, pp.
363
368
.
72.
Johnson
,
T. L.
, and
Rand.
,
R.
1979, “
On the Existence and Bifurcation of Minimal Normal Modes
,”
Int. J. Nonlinear Mech.
,
14
, pp.
1
12
.
73.
Xinhua
,
Z.
, 2004, “
Non-Linear Normal Modes as the Extremal Geodesics of the Riemaniam Manifold
,”
Proceedings of the EUROMECH Colloquium 457 on Non linear modes of vibrating systems
,
Frejus, France
pp.
177
180
.
74.
Anand
,
G. V.
, 1972, “
Natural Modes of a Coupled Nonlinear System, I.
J. Nonlinear Mech.
,
7
, pp.
81
91
.
75.
Yen
,
D.
, 1974, “
On the Normal Modes of Nonlinear Dual-Mass Systems,”
Int. J. Nonlinear Mech.
,
9
, pp.
45
53
.
76.
Vakakis
,
A.
, 1992, “
Non-Similar Normal Oscillations in a Strongly Non-Linear Discrete System
,”
J. Sound Vib.
,
158
, pp.
341
361
.
77.
Happawana
,
G. S.
,
Bajaj
,
A. K.
, and
Azene
,
M.
, 1995, “
An Analytical Solutions to Non-Similar Normal Modes in a Strongly Non-Linear Discrete System
,
J. Sound Vibr.
,
183
(
2
), pp.
361
367
.
78.
Bhattacharyya
,
R.
,
Jain
,
P.
, and
Nair
,
A.
, 2002, “
Normal Mode Localization for a Two Degree-of-Freedom System With Quadratic and Cubic Non-Linearities
,”
J. Sound Vib.
,
249
(
5
), pp.
909
919
.
79.
Mikhlin
,
Y.
, 1997, “
On Non-Linear Normal Vibration Modes that Exist only in an Intermediate Amplitude Range
,”
J. Sound Vib.
,
204
(
1
), pp.
159
161
.
80.
Szemplinska-Stupnickayyt
,
W.
1990,
The Behavior of Nonlinear Vibrating Systems
,
Vols. I and II
,
Kluwer Academic Publishers
,
Dordrecht
.
81.
Pak
,
C. H.
1999,
Nonlinear Normal Mode Dynamics
,
Inha University
,
Seoul, Korea
.
82.
Rand
,
R. H.
, and
Ramani
,
D. V.
, 2001, “
Nonlinear Normal Modes in a System With Nonholonomic Constraints
,”
Nonlinear Dyn.
,
25
, pp.
49
64
.
83.
Pak
,
C. H.
, 2006, “
On the Coupling of Non-Linear Normal Modes
,”
Int. J. Nonlinear Mech.
,
41
, pp.
716
725
.
84.
Wang
,
F.
,
Bajaj
,
A. K.
, and
Kamiya
,
K.
, 2005, “
Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System
,”
Nonlinear Dyn.
,
42
, pp.
233
265
.
85.
Wang
,
F.
, and
Bajaj
,
A. K.
, 2008, “
Nonlinear Normal Modes of an Inertially Coupled Conservative System
,”
J. Vib. Control
14
(
1–2
), pp.
107
134
.
86.
Manevitch
,
L. I.
, 2001, “
The Description of Localized Normal Modes in a Chain of Nonlinear coupled Oscillators using Complex Variables
,”
Nonlinear Dyn.
,
25
, pp.
95
109
.
87.
Morgenthaler
,
G. W.
, 1966, “
Normal Modal Vibrations for Some Damped n-Degree-of-Freedom Nonlinear Systems
,
ASME J. Appl. Mech.
,
33
, pp.
877
880
.
88.
Mikhlin
,
Y.
, 1985, “
The Joining of Local Expansions in the Theory of Nonlinear Oscillations
,”
Prikl. Matem. Mekh.
,
49
(
5
), pp.
738
743
.
89.
Manevitch
,
L. I.
, and
Pinsky
,
M. A.
, 1972, “
On Nonlinear Normal Vibrations in Systems With Two Degrees of Freedom
,”
Prikl. Matem. Mech.
,
8
(
9
), pp.
83
90
.
90.
Manevitch
,
L. I.
, and
Pinsky
,
M. A.
, 1972, “
On the Use of Symmetry for Nonlinear Oscillations Calculation
,”
Izv. AN SSSR MTT
7
(
2
), pp.
43
46
(in Russian).
91.
Ovsiannikov
,
L. S.
1982,
Group Analysis of Differential Equations
,
Academic
,
New York
.
92.
Manevich
,
L. I.
, and
Cherevatskii
,
B. P.
, 1969, “
On the Approximate Determination of Normal Vibrations of Nonlinear Systems With Two Degrees of Freedom
,”
Proceedings of the Dnepropetrovsk Railway Transport Institute: Problems of Strength, Reliability and Destruction of Mechanical Systems
,
Dnepropetrovsk
,
Ukraine,
pp.
45
56
(in Russian).
93.
Mikhlin
,
Y.
, and
Lartseva
,
I.
, 1995, “
Symmetries of Nonlinear Dynamical Models and Normal Vibration Modes
,”
Proceedings of the IMACS Symposium on System Analysis and Simulation
Berlin
[J. Math. Model. Simulation, pp.
207
210
(1995)].
94.
Chechin
,
G. M.
,
Novikova
,
N. V.
, and
Abramenko
,
A. A.
, 2002, “
Bushes of Vibrational Modes for Fermi-Pasta-Ulam Chains
,”
Phys. D
166
, pp.
208
238
.
95.
Chechin
,
M.
, and
Ryabov
,
D. S.
, 2004, “
Three-Dimensional Chaotic Flows With Discrete Symmetries
,”
Phys. Rev. E
69
, pp.
1
9
.
96.
Rand
,
R. H.
, 1973, “
The Geometrical Stability of NNMs in Two DOF Systems
,”
Int. J. Nonlinear Mech.
,
8
, pp.
161
168
.
97.
Pecelli
,
G.
, and
Thomas
,
E. S.
, 1979, “
Normal Modes, Uncoupling, and Stability for a Class of Nonlinear Oscillators
,”
Q. Appl. Math.
,
37
, pp.
281
301
.
98.
Pecelli
,
G.
, and
Thomas
,
E. S.
, 1980, “
Stability of for a Class of Nonlinear Planar Oscillators
,”
Int. J. Nonlinear Mech.
,
15
, pp.
57
70
.
99.
Ince
,
E. L.
, 1926,
Ordinary Differential Equations
,
Lomgman Green
,
London
.
100.
Zhupiev
,
A.
, and
Mikhlin
,
Y.
, 1981, “
Stability and Branching of Normal Modes of Nonlinear Systems
,”
Prikl. Matem. Mekh.
,
45
(
3
), pp.
450
455
.
101.
Mikhlin
,
Y.
,and
Zhupiev
,
A.
, 1997, “
An Application of the Ince Algebraization to the Stability of Nonlinear Normal Vibration Modes
,”
Int. J. Nonlinear Mech.
,
32
(
1
), pp.
493
509
.
102.
Bateman
,
H.
,and
Erdelyi
,
A.
1952,
Higher Transcendental Functions
,
McGraw-Hill
,
New York
.
103.
Zhupiev
,
A.
,and
Mikhlin
,
Y.
, 1984, “
Conditions for Finite Numbers of Instability Domains for Nonlinear Normal Modes
,”
Prikl. Matem. Mekh.
,
48
(
4
), pp.
681
684
.
104.
Dubrovin
,
B. A.
,and
Novikov
,
S. P.
, 1974, “
Periodic and Almost Periodic Analogy of Multi-Soliton Solutions of Korteweg-de-Vries Equation
,”
Sov. Phys. JETP
,
67
(
12
), pp.
2131
2144
.
105.
Novikov
,
S. P.
, 1980,
Theory of Solitons. Inverse Problems Method
,
Nauka, Moscow
.
106.
Pak
,
C. H.
, 1989, “
On the Stability Behavior of Bifurcated Normal Modes in Coupled Nonlinear Systems
,”
ASME J. Appl. Mech.
,
56
, pp.
155
161
.
107.
Rand
,
R. H.
,
Pak
,
C. H.
,and
Vakakis
,
A. F.
, 1992, “
Bifurcation of Nonlinear Normal Modes in a Class of Two Degree of Freedom Systems
,”
Acta Mech.
,
3
, pp.
129
145
.
108.
Pak
,
C. H. R.
,
and
,
R. H.
,
Moon
,
F. C.
, 1992, “
Free Vibrations of a Thin Elastica by Normal Modes
,
Nonlinear Dyn.
,
3
, pp.
347
364
.
109.
Vakakis
,
A. F.
,and
Rand
,
R. H.
,
1992, “
Normal Modes and Global Dynamics of a Two-Degree-of-Freedom Nonlinear System. II. High Energies
,”
Int. J. Nonl. Mech.
,
27
(
5
), pp.
875
888
.
110.
Ng
,
R.
,and
Rand
,
R. H.
, 2003, “
Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation
,”
Nonlinear Dyn.
31
, pp.
73
89
.
111.
Manevich
A. I.
, and
Manevitch
,
L. I.
, 2003, “
Free Oscillations in Conservative and Dissipative Symmetric Cubic Two-Degree-of-Freedom Systems With Closed Natural Frequencies
,”
Meccanica
38
, pp.
335
348
.
112.
Manevich
,
A. I.
,and
Manevitch
,
L. I.
, 2005,
The Mechanics of Nonlinear Systems With Internal Resonances
,
Imperial College
,
London
.
113.
Manevich
,
A. I.
, 2004, “
Normal and Elliptic Modes at Forced Oscillations of Cubic 2DOF Systems With Close Natural Frequencies
,”
Proceedings of the EUROMECH Colloq. 457 on Non Linear Modes of Vibrating Systems
,
Frejus, France
, pp.
33
36
.
114.
Manevich
,
A. I.
, 2007, “
Primary Resonances in Nonlinear Symmetric 2DOF Systems Having Close Natural Frequencies
,”
The Second International Conference of “ Nonlinear Dynamics,”
Kharkov, Ukraine,
pp.
198
203
.
115.
Xinye
,
L.
,
Yushu
,
C.
,and
Zhiqiang
,
W.
, 2004, “
Non-Linear Normal Modes and Their Bifurcation of a Class of Systems With Three Double of Pure Imaginary Roots and Dual Internal Resonances
,”
Int. J. Nonlinear Mech.
,
39
, pp.
189
199
.
116.
Pak
,
C. H.
,and
Choi
,
Y. S.
, 2007, “
On the Sensitivity of Non-generic Bifurcation of Non-Linear Normal Modes
,”
Int. J. Nonlinear Mech.
,
2
, pp.
973
980
.
117.
Synge
,
J. L.
, 1926, “
On the Geometry of Dynamics
,”
Philos. Trans. R. Soc. London Ser. A
226
, pp.
33
106
.
118.
Siegel
,
C. L.
,and
Moser
,
J. K.
, 1971,
Celestial Mechanics (Grundlehren Bd. 187)
,
Spronger
,
Berlin
.
119.
Birkhoff
,
G. D.
1927,
Dynamical Systems,
AMS Publications
,
Providence
.
120.
Whittaker
,
E. T.
, 1959,
A Treatise on the Analytical Dynamics of Particles and Rigid Bodies
,
Cambridge University
,
Cambridge.
121.
Month
,
L. A.
, and
Rand
,
R.H.
, 1980, “
An Application of the Poincare Map to the Stability of Nonlinear Normal Modes
,”
ASME J. Appl. Mech.
,
47
, pp.
645
651
.
122.
Shaw
,
S.
,and
Pierre
,
C.
, 1991, “
Nonlinear Normal Modes and Invariant Manifolds
,”
J. Sound Vib.
,
150
, pp.
170
173
.
123.
Shaw
,
S.
,and
Pierre
,
C.
, 1993, “
Normal Modes for Nonlinear Vibratory Systems
,”
J. Sound Vib.
,
164
, pp.
85
124
.
124.
Guckenheimer
,
J.
,and
Holmes
,
P.
, 1986,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
.
125.
Steindl
,
A.
and
Troger
,
H.
, 2004, “
Invariant Maniforlds in the Dimension Reduction for Dissipative and Conservative Systems
,”
Proceedings of the EUROMECH Colloq. 457 on Nonlinear Modes of Vibrating Systems
,
Frejus, France
pp.
79
82
.
126.
Avramov
,
K. V.
,and
Strel’nikova
,
E. A.
, 2011, “
Nonlinear Normal Modes of Self-Sustained Vibrations of Finite-Degree-of-Freedom Mechanical Systems
,”
Notices of the Nat. Acad. of Scien. of Ukraine
,
2
, pp.
44
51
.
127.
Pesheck
,
E.
,
Boivin
,
N.
,and
Pierre
,
C.
, 2001, “
Nonlinear Modal Analysis of Structural Systems Uusing Multi-Mode Invariant Manifolds
,”
Nonlinear Dyn.
,
25
, pp.
183
205
.
128.
Apiwattanalunggarn
,
P.
,
Shaw
,
S.
,and
Pierre
,
C.
, 2005, “
Component Mode Synthesis Using Nonlinear Normal Modes
,”
Nonlinear Dyn.
,
41
, pp.
17
46
.
129.
Jezequel
,
L.
,and
Lamarque
,
C. H.
, 1991, “
Analysis of Nonlinear Dynamics by the Normal Form Theory
,”
J. Sound Vibr.
,
149
(
3
), pp.
429
452
.
130.
Touze
,
C.
,
Thomas
,
O.
,and
Chaigne
,
A.
, 2004, “
Hardening/ Softening Behaviour in Non-linear Oscillations of Structural Systems Using Non-linear Normal Modes
,”
J. Sound Vib.
,
273
, pp.
77
101
.
131.
Lacarbonara
,
W.
,
Rega
,
G.
,and
Nayfeh
,
A. H.
, 2003, “
Resonant Non-Linear Modes. Part I: Analytical Treatment for Structural One-dimensional Systems
,”
Int. J. Nonlinear Mech.
,
38
, pp.
851
872
.
132.
Lenci
,
S.
,and
Rega
,
G.
, 2008, “
Detecting Stable-unstable Nonlinear Invariant Manifold and Homoclinic Orbits in Mechanical Systems
,
ASME Int. Mech. Eng. Congr. Expos.
IMECE2008–66690, CD-Rom.
133.
Touze
,
C.
,and
Amabili
,
M.
, 2006, “
Nonlinear Normal Modes for Damped Geometrically Nonlinear Systems: Application to Reduced-Order Modeling of Harmonically Forced Structures
,”
J. Sound Vib.
,
298
, pp.
958
981
.
134.
Rosenberg
,
R. M.
, 1966, “
Steady-State Forced Vibrations
,”
Int. J. Nonlinear Mech.
,
1
, pp.
95
108
.
135.
Yang
,
T. L.
,and
Rosenberg
,
R.
, 1968, “
On Forced Vibrations of a Particle in the Plane
,”
Int. J. Nonlinear Mech.
,
3
, pp.
47
63
.
136.
Vakakis
,
A.
,and
Caughey
,
T.
, 1992, “
A Theorem on the Exact Nonsimilar Steady State Motions of a Nonlinear Oscillator
,”
ASME J. Appl. Mech.
,
59
, pp.
418
424
.
137.
Avramov
,
K. V.
, 2008, “
Analysis of Forced Vibrations by Nonlinear Modes
,”
Nonlinear Dyn.
,
53
, pp.
117
127
.
138.
Warminski
,
J.
, 2008, “
Nonlinear Normal Modes of Coupled Self-Excited Oscillators in Regular and Chaotic Vibration Regimes
,”
J. Theor. Appl. Mech.
,
3
, pp.
693
714
.
139.
Rauscher
,
M.
, 1938, “
Steady Oscillations of Systems With Nonlinear and Unsymmetric elasticity
,”
J. Appl. Mech.
,
5
, pp.
A169
177
.
140.
Jiang
,
D.
,
Pierre
,
C.
, and
Shaw
,
S. W.
, 2005, “
Nonlinear Normal Modes for Vibratory Systems Under Harmonic Excitation
,”
J. Sound Vib.
,
288
(
4–5
), pp.
791
812
.
141.
Zhuravlev
,
V. P.
, 1992, “
On Special Directions in Configuration Space of Linear Vibrating systems
,”
Prikl. Matem. Mekh.
,
1
, pp.
16
23
.
142.
Pilipchuk
,
V. N.
, 2000, “
Principal Trajectories of the Forced Vibration for Discrete and Continuous Systems
,”
Meccanica
35
(
6
), pp.
497
517
.
143.
Gendelman
,
O. V.
, 2008, “
Nonlinear Normal Modes in Homogeneous System With Time Delays
,”
Nonlinear Dyn.
,
52
, pp.
367
376
.
144.
Seydel
,
R.
, 1997, “
Nonlinear Computation
,”
Int. J. Bifurcation Chaos Appl. Sci. Eng.
,
7
, pp.
2105
2126
.
145.
Slater
,
J. C.
, 1996, “
A Numerical Method for Determining Nonlinear Normal Modes
,
Nonlinear Dyn.
,
10
, pp.
19
30
.
146.
Burton
,
T. D.
, 2007, “
Numerical Calculations of Nonlinear Normal Modes in Structural Systems
,”
Nonlinear Dyn.
,
49
, pp.
425
441
.
147.
Arquier
,
R
,
Perignon
,
F.
,and
Cochelin
,
B.
, 2004, “
Numerical Continuation of Non Linear Modes of Elastic Structures
,”
Proceedings of the EUROMECH Colloquium 457 on Non Linear Modes of Vibrating Systems
,
Frejus, France
, pp.
17
20
.
148.
Nguyen
,
T. M.
,
Argoul
,
P.
,and
Bonnet
,
G.
, 2006, “
Comparison of Three Methods for Constructing Nonlinear Normal Modes
,”
2nd International Conference on Nonlinear Normal Modes and Localization in Vibratory Systems, Book of abstracts
,
Samos, Greece
,
57
58
.
149.
Lewandowski
,
R.
, 1997, “
Computational Formulation for Periodic Vibrations of Geometrically Nonlinear Structures. Part 1: Theoretical Background
,”
Int. J. Solids Struct.
,
34
(
15
), pp.
1925
1947
;
Lewandowski
,
R.
, “
Computational Formulation for Periodic Vibrations of Geometrically Nonlinear Structures. Part 2: Numerical Strategy and Examples
,”
ibid.
34
(
15
), pp.
1949
1964
.
150.
Mikhlin
,
Y. V.
,
Shmatko
,
T. V.
,and
Manucharyan
,
G. V.
, 2004, “
Lyapunov Definition and Stability of Regular or Chaotic Vibration Modes in Systems With Several Equilibrium Positions
,”
Computers and Structures
,
82
, pp.
2733
2742
.
151.
Mikhlin
,
Y. V.
,
Vakakis
,
A. F.
,and
Salenger
,
G.
, 1998, “
Direct and Inverse Problems Encountered in Vibro-Impact Oscillations of a Discrete System
,”
J. Sound Vib.
,
216
(
2
), pp.
227
250
.
152.
Aziz
,
M. A.
,
Vakakis
,
A. F.
,and
Manevich
,
L. I.
, 1999, “
Exact Solutions of the Problem of the Vibro-Impact Oscillations of a Discrete System With Two Degrees of Freedom
,”
J. Appl. Math. Mech.
,
63
(
4
), pp.
527
530
.
153.
Pilipchuk
,
V. N.
, 2009, “
Impact Mode Superpositions and Parameter Variations
,”
Vibro-Impact Dynamics of Ocean Systems
(LNACM 44),
R. A.
Ibrahim
,
V. I.
Babitsky
, and
M.
Okuma
, eds.,
Springer-Verlag
,
Berlin
, pp.
231
243
.
154.
Avramov
,
K. V.
,and
Borysiuk
,
O. V.
, 2008, “
Analysis of an Impact Duffing Oscillator by Means of a Nonsmooth Unfolding Transformation
,”
J. Sound Vib.
,
318
, pp.
1197
1209
.
155.
Pilipchuk
,
V. N.
, 2006, “
A Periodic Version of Lie Series for Normal Mode Dynamics
,”
Nonlinear Dyn. Sys. Th.
,
6
(
2
), pp.
187
190
.
156.
Vestroni
,
F.
,
Luongo
,
A.
,and
Paolone
,
A.
, 2008, “
A Perturbation Method for Evaluating Nonlinear Normal Modes of a Piecewise Linear 2-DOF System
,”
Nonlinear Dyn.
,
54
, pp.
379
393
.
157.
Chen
,
S. L.
,and
Shaw
,
S. W.
, 1996, “
Normal Modes for Piecewise Linear Vibratory Systems
,”
Nonlinear Dyn.
,
10
, pp.
135
164
.
158.
Jiang
,
D.
,
Pierre
,
C.
,and
Shaw
,
S. W.
, 2004, “
Large-Amplitude Non-Linear Normal Modes of Piecewise Linear Systems
,”
J. Sound Vib.
,
272
, pp.
869
891
.
159.
Boivin
,
N.
,
Pierre
,
C.
,and
Shaw
,
S. W.
, 1995, “
Non-Linear Normal Modes, Invariance and Modal Dynamics Approximations on Non-Linear Systems
,”
Nonlinear Dyn.
,
8
, pp.
315
346
.
160.
Szemplinska-Stupnicka
,
W.
, 1983, “
Nonlinear Normal Modes” and the Generalized Ritz Method in the Problems of Vibrations of Nonlinear Elastic Continuous Systems
,”
Int. J. Nonlinear Mech.
,
18
(
2
), pp.
149
165
.
161.
King
,
M. E.
,and
Vakakis
,
A. F.
, 1995, “
Asymptotic Analysis of Nonlinear Mode Localization in a Class of Coupled Continuous Structures
,”
Int. J. Solid. Struc.
,
32
(
8/9
), pp.
1161
1177
.
162.
Nayfeh
,
A. H.
,
Chin
,
C.
,and
Nayfeh
,
S. A.
, 1996, “
On Nonlinear Normal Modes of Systems With Internal Resonance
,”
ASME J. Vib. Acostic.
,
118
, pp.
340
345
.
163.
Nayfeh
,
A. H.
2000,
Nonlinear Interactions
,
Wiley
,
New York
.
164.
Nayfeh
,
A. H.
,and
Nayfeh
,
S. A.
, 1995, “
Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities
,”
ASME J. Vib. Acostic.
,
117
, pp.
199
206
.
165.
Andrianov
,
I. V.
,and
Kholod
,
E. G.
, 1993, “
Intermediate Asymptotical Forms in Nonlinear Dynamics of Shells
,”
Mech. Sol.
,
28
(
2
), pp.
160
165
.
166.
Barenblatt
,
G. I.
,and
Zel’dovitch
,
Y. a.
, 1971, “
Intermediate Asymptotics in Mathematical Physics
,”
Russ. Math. Surv.
,
26
(
2
), pp.
115
129
.
167.
Bolotin
,
V. V.
, 1961, “
An Asymptotic Method for the Study of the Problem of Eigenvalues of Rectangular Regions
,”
Problems of Continuum Mechanics
,
SIAM
,
Philadelphia
, pp.
56
58
.
168.
Andrianov
,
I. V.
,and
Kholod
,
E. G.
, 1993, “
Non-linear Free Vibration of Shallow Cylindrical Shell by Bolotin’s Asymptotic Method
, “
J. Sound Vib.
,
165
(
1
), pp.
9
17
.
169.
Andrianov
,
I. V.
,and
Krizhevsky
,
G. A.
, 1993, “
Free Vibration Analysis of Rectangular Plates With Structural Inhomogenity
,”
J. Sound Vib.
,
162
(
2
), pp.
231
241
.
170.
Andrianov
,
I. V.
,and
Danishevskiy
,
V. V.
, 1995, “
Asymptotic Investigation of the Nonlinear Dynamic Boundary Value Problem for Rod
,”
Tech. Mech.
,
15
(
1
), pp.
53
55
.
171.
Andrianov
,
I. V.
,and
Danishevs’kyy
,
V. V.
, 2002, “
Asymptotic Approach for Non–Linear Periodical Vibrations of Continuous Structures
,”
J. Sound Vib.
,
249
(
3
), pp.
465
481
.
172.
Andrianov
,
I. V.
, 2008, “
Asymptotic Construction of Nonlinear Normal Modes for Continuous Systems
,”
Nonlinear Dyn.
,
51
(
1/2
), pp.
99
109
.
173.
Pierre
,
C.
,and
Dowell
,
E. H.
, 1987, “
Localization of Vibration by Structural Irregularity
,”
J. Sound Vib.
,
114
(
3
), pp.
411
424
.
174.
Bendiksen
,
O. O.
, 2000, “
Localization Phenomena in Structural Dynamics
,”
Chaos, Solitons Fractals
11
, pp.
1621
1660
175.
Vakakis
,
A. F.
,and
Cetinkaya
,
C.
, 1993, “
Mode Localization in a Class of Multi-Degree-of-Freedom Systems With Cyclic Symmetry
,”
SIAM J. Appl. Math.
,
53
, pp.
265
282
.
176.
Vakakis
,
A. F.
,
Nayfeh
,
T.
,and
King
,
M. E.
, 1993, “
A Multiple-Scales Analysis of Nonlinear, Localized Modes in Cyclic Periodic System
,”
ASME J. Appl. Mech.
,
60
(
2
), pp.
388
397
.
177.
Vakakis
,
A. F.
, 1993, “
Passive Spatial Confinement of Impulsive Excitations in Coupled Nonlinear Beams
,”
AIAA J.
,
32
(
9
), pp.
1902
1910
.
178.
King
,
M. E.
,and
Vakakis
,
A. F.
, 1995, “
Mode Localization in a System of Coupled Flexible Beams With Geometric Nonlinearities
,”
ZAMM
75
(
2
), pp.
127
139
.
179.
King
,
M. E.
,and
Layne
,
P. A.
, 1998, “
Dynamics of Nonlinear Cyclic Systems With Structural Irregularity
,”
Nonlinear Dyn.
,
15
, pp.
225
244
.
180.
Cai
,
C. W.
,
Chan
,
H. C.
,and
Cheung
,
Y. K.
, 1997, “
Localized Modes in Periodic Systems With Nonlinear Disorders
,”
ASME J. Appl. Mech.
,
64
, pp.
940
945
.
181.
Cai
,
C. W.
,
Chan
,
H. C.
,
Cheung
,
and Y. K.
, 2000, “
Localized Modes in a Two-Degree-Coupling Periodic System With a Nonlinear Disordered Subsystem
,”
Chaos, Solitons Fractals
11
, pp.
1481
1492
.
182.
Vakakis
,
A. F.
,
King
,
M. E.
,and
Pearlstrin
,
A. J.
, 1994, “
Forced Localization in a Periodic Chain of Nonlinear Oscillators
,”
Int. J. Nonlinear Mech.
,
29
, pp.
429
447
.
183.
Weinstein
,
A.
, 1973, “
Normal Modes for Nonlinear Hamiltonian Systems
,”
Inv. Math.
,
20
, pp.
47
57
.
184.
Moser
,
J. K.
, 1976, “
Periodic Orbits Near an Equilibrium and a Theorem by Alan Weinstein
,”
Comm. Pur. Appl. Math.
,
29
, pp.
727
747
.
185.
Weinstein
,
A.
, 1978, “
Periodic Orbits for Convex Hamiltonian Systems
,
Ann. Math.
,
108
, pp.
507
518
.
186.
Ekeland
,
I.
,and
Lasry
,
J. M.
, 1980, “
On the Number of Periodic Trajectories for a Hamiltonian Flow on a Convex Energy Surface
,”
Ann. Math.
112
, pp.
283
319
.
187.
Rabinowitz
,
P. H.
, 1982, “
On Large Norm Periodic Solutions of Some Differential Equations
,”
Ergod. Theory Dyn. Syst.
,
11
, pp.
193
210
.
188.
Cook
,
C.
,and
Struble
,
R.
, 1966, “
On the Existence of Periodic Solutions and Normal Mode Vibrations of Nonlinear Systems
,”
Quart. Appl. Math.
,
24
(
3
), pp.
177
193
.
189.
Cook
,
C.
,and
Struble
,
R. A.
, 1966, “
Perturbations of Normal Mode Vibrations
,”
Int. J. Nonl. Mech.
,
1
(
2
), pp.
147
155
.
190.
Pak
,
C. H.
,and
Rosenberg
,
R. M.
, 1968, “
On the Existence of Normal Mode Vibrations of Nonlinear Systems
,”
Quart. Appl. Math
.
26
, pp.
403
416
.
191.
Zevin
,
A. A.
, 1986, “
Non-Local Criteria for the Existence and Stability of Periodic Oscillations in Autonomous Hamiltonian Systems
,”
Prikl. Matem. Mekh.
,
50
(
1
), pp.
64
72
.
192.
Zevin
,
A. A.
, 1992, “
Qualitative Analysis of Periodic Oscillations in Autonomous Hamiltonian Systems
,”
Int. J. Nonlinear Mech.
,
28
(
3
), pp.
281
290
.
You do not currently have access to this content.