A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

1.
Benamar
R.
,
Bennouna
M. M. K.
, and
White
R. G.
,
1991
, “
The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, Part I: Simply supported and clamped-clamped beams
,”
Journal of Sound and Vibration
, Vol.
149
, pp.
179
195
.
2.
Bennouna
M. M. K.
, and
White
R. G.
,
1984
, “
The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam
,”
Journal of Sound and Vibration
, Vol.
96
, pp.
309
331
.
3.
Boivin, N., Pierre, C., and Shaw, S. W., 1993, “Nonlinear normal modes, invariance, and modal dynamics approximations of nonlinear systems,” Proceedings of the 14th Biennial Conference on Mechanical Vibration and Noise, Albuquerque, NM, Sept. 19–22.
4.
Caughey
T. K.
,
Vakakis
A. F.
, and
Sivo
J.
,
1990
, “
Analytical study of similar normal modes and their bifurcations in a class of strongly nonlinear systems
,”
International Journal of Non-Linear Mechanics
, Vol.
25
, No.
5
, pp.
521
533
.
5.
Cooke
C. H.
, and
Struble
R. A.
,
1966
, “
On the existence of periodic solutions and normal mode vibrations of nonlinear systems
,”
Quarterly of Applied Mathematics
, Vol.
24
, No.
3
, pp.
177
193
.
6.
King, M. E., 1995, “Analytical and experimental aspects of nonlinear normal modes and nonlinear mode localization,” Ph.D. Thesis, University of Illinois, Urbana, IL.
7.
King
M. E.
, and
Vakakis
A. F.
,
1994
, “
An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems
,”
ASME Journal of Vibration and Acoustics
, Vol.
116
, pp.
332
340
.
8.
King
M. E.
, and
Vakakis
A. F.
,
1995
, “
Asymptotic analysis of nonlinear mode localization in a class of coupled continuous structures
,”
Int. Journal of Solids and Structures
, Vol.
32
, No.
8–9
, pp.
1161
1177
.
9.
Manevitch
L. I.
, and
Mikhlin
Y. V.
,
1972
, “
On periodic solutions close to rectilinear normal vibration modes
,”
PMM
, Vol.
36
, No.
6
, pp.
1051
1058
.
10.
Mikhlin
Y. V.
,
1974
, “
Resonance modes of near-conservative nonlinear systems
,”
PMM
, Vol.
38
, No.
3
, pp.
459
464
.
11.
Month
L. A.
, and
Rand
R. H.
,
1977
, “
The Stability of Bifurcating Periodic Solutions in a Two Degree-of-Freedom Nonlinear System
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
44
, pp.
782
783
.
12.
Nayfeh, A. H., and Chin, C., 1994, “On Nonlinear Normal Modes of Systems With Internal Resonance,” ASME Journal of Vibration and Acoustics, in press.
13.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.
14.
Nayfeh
A. H.
, and
Nayfeh
S. A.
,
1994
, “
On Nonlinear Modes of Continuous Systems
,”
ASME Journal of Vibration and Acoustics
, Vol.
116
, pp.
129
136
.
15.
Rand
R. H.
,
1971
, “
A higher order approximation for Nonlinear Normal Modes in two-dof Systems
,”
International Journal of Non-Linear Mechanics
, Vol.
6
, pp.
545
547
.
16.
Rand, R. H., 1995, personal communications with authors.
17.
Rand
R. H.
,
Pak
C. H.
, and
Vakakis
A. F.
,
1992
, “
Bifurcations of nonlinear normal modes in a class of two degree of freedom systems
,”
Acta Mechanica
, [Suppl] Vol.
3
, pp.
129
145
.
18.
Rosenberg
R. M.
,
1964
, “
On the existence of normal mode vibrations of nonlinear systems with two degrees of freedom
,”
Quarterly of Applied Mathematics
, Vol.
22
, No.
3
, pp.
217
234
.
19.
Rosenberg
R. M.
,
1966
, “
On non-linear vibrations of systems with many degrees of freedom
,”
Advances in Applied Mechanics
, Vol.
9
, pp.
155
242
.
20.
Shaw
S. W.
, and
Pierre
C.
,
1991
, “
Nonlinear normal modes and invariant manifolds
,”
Journal of Sound and Vibration
, Vol.
150
, pp.
170
173
.
21.
Shaw, S. W., and Pierre, C., 1992, “On nonlinear normal modes,” Winter Annual Meeting, ASME, Anaheim, CA, Nov. 8–13.
22.
Shaw
S. W.
, and
Pierre
C.
,
1993
, “
Normal modes for nonlinear vibratory systems
,”
Journal of Sound and Vibration
, Vol.
164
, No.
1
, pp.
85
96
.
23.
Shaw
S. W.
, and
Pierre
C.
,
1994
, “
Normal modes of vibration for nonlinear continuous systems
,”
Journal of Sound and Vibration
, Vol.
169
, No.
3
, pp.
319
348
.
24.
Vakakis, A. F., 1990, “Analysis and identification of linear and nonlinear normal modes in vibrating systems,” Ph.D. Thesis, California Institute of Technology, Pasadena, CA.
25.
Vakakis
A. F.
,
1992
a, “
Fundamental and subharmonic resonances in a system with a ‘1–1’ internal resonance
,”
Nonlinear Dynamics
, Vol.
3
, pp.
123
143
.
26.
Vakakis
A. F.
,
1992
b, “
Nonsimilar normal oscillations in a strongly nonlinear discrete system
,”
Journal of Sound and Vibration
, Vol.
151
, No.
22
, pp.
341
361
.
27.
Vakakis
A. F.
, and
Caughey
T. K.
,
1992
, “
A Theorem on the Exact Nonsimilar Steady-State Motions of a Nonlinear Oscillator
,”
ASME JOURNAL OF APPLIED MECHANICS
, Vol.
59
, pp.
418
424
.
28.
Vakakis, A. F., Manevitch, L. I., Mikhlin, Y. V., Pilipchuk, V. N., and Zevin, A. A., 1995, Normal Modes and Localization in Continuous Systems, Wiley Interscience, in press.
29.
Weinstein
A.
,
1973
, “
Normal modes of nonlinear hamiltonian systems
,”
Inventiones math.
, Vol.
20
, pp.
47
57
.
This content is only available via PDF.
You do not currently have access to this content.