The present work is concerned with deterministic nonlinear phenomena arising in the finite-amplitude dynamics of elastic suspended cables. The underlying theoretical framework has been addressed in Part I of this Special Issue, where the mechanical system and its mathematical modeling have been presented, and different techniques for the analysis of nonlinear dynamics have been illustrated with reference to the suspended cable. Herein, we discuss the main features of system regular and complex response, and the associated bifurcational behavior. Nonlinear phenomena are considered separately for single-degree-of-freedom and multidegree-of-freedom cable models, by distinguishing between theoretical and experimental results and comparing them with each other. Regular and nonregular vibrations are considered either in the absence of internal resonance or under various internal/external, and possibly simultaneous, resonance conditions. The most robust classes of steady periodic motions, the relevant response scenarios in control parameter space, and the main features of multimodal interaction phenomena are summarized. Bifurcation and chaos phenomena are discussed for the single-dof model by analyzing the local and global features of steady nonregular dynamics. For the experimental model, the most meaningful scenarios of transition to chaos are illustrated, together with the properties of the ensuing quasiperiodic and chaotic attractors. Finally, the important issues of determining system dimensionality and identifying properly reduced-order theoretical models of cable are addressed. There are 185 references listed in this review article.
Issue Section:
Review Articles
Keywords:
bifurcation,
reviews,
finite element analysis,
nonlinear dynamical systems,
chaos,
elasticity
Topics:
Attractors,
Bifurcation,
Cables,
Chaos,
Dynamics (Mechanics),
Resonance,
Vibration,
Excitation
1.
Rega
G
(2004
), Nonlinear vibrations of suspended cables, Part I: Modeling and analysis
, Appl. Mech. Rev.
57
(6
), 443
–478
.2.
Carrier
GF
(1945
), On the nonlinear vibration problem of the elastic string
, Q. Appl. Math.
3
, 157
–165
.3.
Anand
GV
(1966
), Nonlinear resonance in stretched strings with viscous damping
, J. Acoust. Soc. Am.
40
, 1517
–1528
.4.
Anand
GV
(1969
), Stability of nonlinear oscillations of a stretched string
, J. Acoust. Soc. Am.
46
, 667
–677
.5.
Narashima
R
(1968
), Nonlinear vibration of an elastic string
, J. Sound Vib.
8
, 134
–146
.6.
Antman
SS
(1980
), The equations for large vibrations of strings
, Am. Math. Monthly
87
, 359
–370
.7.
Miles
J
(1984
), Resonant, nonplanar motion of a stretched string
, J. Acoust. Soc. Am.
75
, 1505
–1510
.8.
Tufillaro
NB
(1989
), Nonlinear and chaotic string vibrations
, Am. J. Phys.
57
, 408
–414
.9.
Johnson
JM
, and Bajaj
AK
(1989
), Amplitude modulated and chaotic dynamics in resonant motion of strings
, J. Sound Vib.
128
, 87
–107
.10.
O’Reilly
O
, and Holmes
PJ
(1992
), Non-linear, non-planar and non-periodic vibrations of a string
, J. Sound Vib.
153
, 413
–435
.11.
Leissa
AW
, and Saad
AM
(1994
), Large amplitude vibrations of strings
, ASME J. Appl. Mech.
61
, 296
–301
.12.
Nayfeh
SA
, Nayfeh
AH
, and Mook
DT
(1995
), Nonlinear response of a taut string to longitudinal and transverse end excitation
, J. Vib. Control
1
, 307
–334
.13.
Rubin
MB
, and Gottlieb
O
(1996
), Numerical solutions of forced vibration and whirling of a nonlinear string using the theory of a Cosserat point
, J. Sound Vib.
197
, 85
–101
.14.
Nayfeh AH, and Mook DT (1979), Nonlinear Oscillations, Wiley, New York.
15.
Luongo
A
, Rega
G
, and Vestroni
F
(1984
), Planar non-linear free vibrations of an elastic cable
, Int. J. Non-Linear Mech.
19
, 39
–52
.16.
Rega
G
, Vestroni
F
, and Benedettini
F
(1984
), Parametric analysis of large-amplitude free vibrations of a suspended cable
, Int. J. Solids Struct.
20
, 95
–105
.17.
Takahashi
K
, and Konishi
Y
(1987a
), Non-linear vibrations of cables in three-dimensions, Part I: Nonlinear free vibrations
, J. Sound Vib.
118
, 69
–84
.18.
Irvine
HM
, and Caughey
TK
(1974
), The linear theory of free vibrations of a suspended cable
, Proc. R. Soc. London, Ser. A
341
, 299
–315
.19.
Takahashi
K
, Wu
Q
, and Nakamura
S
(2003
), In-plane antisymmetric response of cables through bifurcation under symmetric sinusoidally time-varying load
, J. Sound Vib.
268
, 1
–14
.20.
Rega G, Lacarbonara W, and Nayfeh AH (2000), Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature, IUTAM Symp Recent Developments in Nonlinear Oscillations of Mechanical Systems, E Kreuzer and N van Dao (eds), Solid Mechanics and Its Applications, 77, 235–246, Kluwer, Dordrecht.
21.
Lacarbonara
W
(1999
), Direct treatment and discretizations of nonlinear spatially continuous systems
, J. Sound Vib.
221
, 849
–866
.22.
Arafat
HN
, and Nayfeh
AH
(2003
), Nonlinear responses of suspended cables to primary resonance excitations
, J. Sound Vib.
266
, 325
–354
.23.
Gattulli
V
, Martinelli
L
, Perotti
F
, and Vestroni
F
(2004
), Nonlinear oscillations of cables under harmonic loading using analytical and finite element models
, Comput. Methods Appl. Mech. Eng.
193
, 69
–85
.24.
Desai
YM
, Popplewell
N
, Shah
A
, and Buragohain
DN
(1988
), Geometric nonlinear analysis of cable supported structures
, Comput. Struct.
29
, 1001
–1009
.25.
Martinelli
L
, and Perotti
F
(2001
), Numerical analysis of the nonlinear dynamic behaviour of suspended cables under turbulent wind excitation
, Int. J. Struct. Stab. Dyn.
1
, 207
–233
.26.
Rienstra SW (1988), A nonlinear theory of free vibrations of single and coupled suspended elastic cables, Report WD 88-06, Katholieke Univ Nijmegen, The Netherlands.
27.
Rienstra SW (1990), Nonlinear free vibrations of coupled spans of suspended cables, Proc. 3rd Eur. Conf. Mathematics in Industry, J Manley et al. (eds), Kluwer and Teubner, Stuttgart, 133–144.
28.
Rawlins
CB
(2004
), Effect of non-linearity in free large oscillations of a shallow catenary
, J. Sound Vib.
273
, 857
–874
.29.
Irvine HM (1981), Cable Structures, MIT Press (1992, Dover Publ, New York).
30.
Benedettini
F
, and Rega
G
(1987
), Nonlinear dynamics of an elastic cable under planar excitation
, Int. J. Non-Linear Mech.
22
, 497
–509
.31.
Gattulli
V
, Pasca
M
, and Vestroni
F
(1997
), Nonlinear oscillations of a nonresonant cable under in-plane excitation with a longitudinal control
, Nonlinear Dyn.
14
, 139
–156
.32.
Benedettini
F
, and Rega
G
(1989
), Planar non-linear oscillations of elastic cables under superharmonic resonance conditions
, J. Sound Vib.
132
, 353
–366
.33.
Benedettini
F
, Rega
G
, and Salvatori
A
(1992
), Prediction of bifurcation and chaos for an asymmetric elastic oscillator
, Chaos, Solitons Fractals
2
, 303
–321
.34.
Rega
G
, and Benedettini
F
(1989
), Planar non-linear oscillations of elastic cables under subharmonic resonance conditions
, J. Sound Vib.
132
, 367
–381
.35.
Rega G (1995), Bifurcation and chaos in the Helmholtz-Duffing oscillator, Bifurcation and Chaos, J Awrejcewicz (ed), Springer-Verlag, Berlin, 191–215.
36.
Szemplinska-Stupnicka
W
, and Bajkowski
J
(1986
), The 1/2-subharmonic resonance and its transition to chaotic motion in a nonlinear oscillator
, Int. J. Non-Linear Mech.
21
, 401
–419
.37.
Rega
G
, Benedettini
F
, and Salvatori
A
(1991
), Periodic and chaotic motions of an unsymmetric oscillator in nonlinear structural dynamics
, Chaos, Solitons Fractals
1
, 39
–54
.38.
Benedettini
F
, and Rega
G
(1990
), Numerical simulations of chaotic dynamics in a model of elastic cable
, Nonlinear Dyn.
1
, 23
–38
.39.
Moon FC (1992), Chaotic and Fractal Dynamics, Wiley, New York.
40.
Parker TS, and Chua LO (1989), Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York.
41.
Hsu
CS
(1992
), Global analysis by cell mapping
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
2
, 727
–771
.42.
Seydel R (1994), Practical Bifurcation and Stability Analysis, Springer-Verlag, New York.
43.
Guckenheimer J, and Holmes P (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York.
44.
Wiggins S (1990), Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York.
45.
Abraham RH, and Shaw CD (1992), Dynamics: The Geometry of Behaviour, Addison-Wesley, Redwood City.
46.
Rega
G
, Salvatori
A
, and Benedettini
F
(1995
), Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator
, Nonlinear Dyn.
7
, 249
–272
.47.
Rega
G
, and Salvatori
A
(1996a
), Bifurcation structure at 1/3-subharmonic resonance in an asymmetric nonlinear elastic oscillator
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
6
, 1529
–1546
.48.
Ueda
Y
(1991
), Survey of regular and chaotic phenomena in the forced Duffing oscillator
, Chaos, Solitons Fractals
1
, 199
–231
.49.
Grebogi
C
, Ott
E
, and Yorke
JA
(1983
), Crises, sudden changes in chaotic attractors, and transient chaos
, Physica D
7
, 181
–200
.50.
Grebogi
C
, Ott
E
, and Yorke
JA
(1987
), Basin boundary metamorphoses: Changes in accessible boundary orbits
, Physica D
24
, 243
–262
.51.
Kleczka M, Kreuzer E, and Wilmers C (1989), Crises in mechanical systems, IUTAM Symp Nonlinear Dynamics in Engineering Systems, W Schiehlen (ed), Springer-Verlag, Berlin-Heidelberg, 141–148.
52.
Soliman
MS
, and Thompson
JMT
(1991
), Basin organization prior to a tangled saddle-node bifurcation
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
1
, 107
–118
.53.
Sommerer
JC
, and Grebogi
C
(1992
), Determination of crisis parameter values by direct observation of manifold tangencies
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
2
, 383
–396
.54.
Katz
AL
, and Dowell
EH
(1994
), From single well chaos to cross well chaos: A detailed explanation in terms of manifold intersections
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
4
, 933
–941
.55.
Rega G, and Salvatori A (1996b), Attractor-basin sudden bifurcation mechanisms in nolinear structural dynamics, Proc. Int. Conf. Nonlinearity, Bifurcation and Chaos, J Awrejcewicz and CH Lamarque (eds), Lodz, 222–225.
56.
Lai YC, Grebogi C, and Yorke JA (1992), Sudden change in the size of chaotic attractors: How does it occur? Applied Chaos, JH Kim and J Stringer (eds), Wiley, New York, 441–455.
57.
Salvatori A, and Rega G (1996), Computational characterization of attractor-basin structure in the nonlinear dynamics of multidegree-of-freedom structural models, Structural Dynamics, G Augusti, C Borri, and P Spinelli (eds), Balkema, Rotterdam, 341–348.
58.
Nayfeh
AH
, and Balachandran
B
(1989
), Modal interactions in dynamical and structural systems
, Appl. Mech. Rev.
42
, S175–S201
S175–S201
.59.
Nayfeh AH (2000), Nonlinear Interactions. Analytical, Computational, and Experimental Methods, Wiley Series in Nonlinear Science, New York.
60.
Lacarbonara
W
, Rega
G
, and Nayfeh
AH
(2003
), Resonant nonlinear normal modes of shallow one-dimensional structural systems, Part I: Analytical treatment
, Int. J. Non-Linear Mech.
38
, 851
–872
.61.
Lacarbonara
W
, and Rega
G
(2003
), Resonant nonlinear normal modes of shallow one-dimensional structural systems, Part II: Activation/orthogonality conditions
, Int. J. Non-Linear Mech.
38
, 873
–887
.62.
Srinil
N
, Rega
G
, and Chucheepsakul
S
(2004
), Three-dimensional nonlinear coupling and dynamic tension in the large amplitude free vibrations of arbitrarily sagged cables
, J. Sound Vib.
269
, 823
–852
.63.
Benedettini
F
, Rega
G
, and Vestroni
F
(1986
), Modal coupling in the free nonplanar finite motion of an elastic cable
, Meccanica
21
, 38
–46
.64.
Luongo
A
, Rega
G
, and Vestroni
F
(1982
), Monofrequent oscillations of a nonlinear model of suspended cable
, J. Sound Vib.
82
, 247
–259
.65.
Yamaguchi H, Miyata T, and Ito M (1978), A behaviour on nonlinear dynamic response of cable systems, Proc. 24th Symp. Structural Engineering, 55–61.
66.
Takahashi
K
, and Konishi
Y
(1987b
), Non-linear vibrations of cables in three-dimensions, Part II: Out-of-plane vibration under in-plane sinusoidally time-varying load
, J. Sound Vib.
118
, 85
–97
.67.
Iyengar
RN
, and Rao
GVV
(1988
), Free vibrations and parametric instability of a laterally loaded cable
, J. Sound Vib.
127
, 231
–243
.68.
Pasca
M
, Vestroni
F
, and Gattulli
V
(1998
), Active longitudinal control of wind-induced oscillations of a suspended cable
, Meccanica
33
, 255
–266
.69.
Zhao
YY
, Wang
LH
, Chen
DL
, and Jiang
LZ
(2002
), Non-linear dynamic analysis of the two-dimensional simplified model of an elastic cable
, J. Sound Vib.
255
, 43
–59
.70.
El-Attar
M
, Ghobarah
A
, and Aziz
TS
(2000
), Non-linear cable response to multiple support periodic excitation
, Eng. Struct.
22
, 1301
–1312
.71.
Tadjbakhsh
IG
, and Wang
Y
(1990
), Wind-driven nonlinear oscillations of cables
, Nonlinear Dyn.
1
, 265
–291
.72.
Cardona
A
, Lerusse
A
, and Ge´radin
M
(1998
), Fast Fourier nonlinear vibration analysis
, Comput. Mech.
22
, 128
–142
.73.
Al-Noury
SI
, and Ali
SA
(1985
), Large-amplitude vibrations of parabolic cables
, J. Sound Vib.
101
, 451
–462
.74.
Luongo
A
, Rega
G
, and Vestroni
F
(1987
), On large-amplitude vibrations of cables
, J. Sound Vib.
116
, 573
–575
.75.
Luongo
A
, Paolone
A
, and Piccardo
G
(1998
), Postcritical behaviour of cables undergoing two simultaneous galloping modes
, Meccanica
33
, 229
–242
.76.
Nielsen
SRK
, and Kirkegaard
PH
(2002
), Super and combinatorial harmonic response of flexible elastic cables with small sag
, J. Sound Vib.
251
, 79
–102
.77.
Berlioz A, and Lamarque CH (2005), Nonlinear vibrations of an inclined cable, J. Vibr. Acoust. (accepted).
78.
Xu
YL
, and Yu
Z
(1999
), Non-linear vibration of cable-damper systems. Part II: Application and verification
, J. Sound Vib.
225
, 465
–481
.79.
Triantafyllou
MS
(1984
), The dynamics of taut inclined cables
, Q. J. Mech. Appl. Math.
37
, 421
–440
.80.
Pakdemirli
M
, Nayfeh
SA
, and Nayfeh
AH
(1995
), Analysis of one-to-one autoparametric resonances in cables—Discretization vs direct treatment
, Nonlinear Dyn.
8
, 65
–83
.81.
Perkins
NC
(1992
), Modal interactions in the non-linear response of elastic cables under parametric/external excitation
, Int. J. Non-Linear Mech.
27
, 233
–250
.82.
Zhang
W
, and Tang
Y
(2002
), Global dynamics of the cable under combined parametrical and external excitations
, Int. J. Non-Linear Mech.
37
, 505
–526
.83.
Rao
GVV
, and Iyengar
RN
(1991
), Internal resonance and nonlinear response of a cable under periodic excitation
, J. Sound Vib.
149
, 25
–41
.84.
Lee
CL
, and Perkins
NC
(1992
), Nonlinear oscillations of suspended cables containing a two-to-one internal resonance
, Nonlinear Dyn.
3
, 465
–490
.85.
Savovich MK, and Ivovich VA (2003), Analysis of two-to-one autoparametric vibration in cable structures: analytical development and numerical treatment, Proc. 5th Int. Symp. Cable Dynamics, Santa Margherita Ligure, AIM, Lie`ge, 205–212.
86.
Chang
WK
, Ibrahim
RA
, and Afaneh
AA
(1996
), Planar and nonplanar nonlinear dynamics of suspended cables under random in-plane loading, Part I: Single internal resonance
, Int. J. Non-Linear Mech.
31
, 837
–859
.87.
Luongo
A
, and Piccardo
G
(1998
), Nonlinear galloping of sagged cables in 1:2 internal resonance
, J. Sound Vib.
214
, 915
–940
.88.
Lee CL, and Perkins NC (1993), Experimental investigation of isolated and simultaneous internal resonances in suspended cables, Nonlinear Vibration ASME, DE-54, 21–31.
89.
Benedettini F, Rega G, and Alaggio R (1996), Experimental analysis of the finite dynamics of a suspended cable, Vibration of Nonlinear, Random, and Time-Varying Systems ASME, DE-84-1, 543–552.
90.
Rega
G
, Alaggio
R
, and Benedettini
F
(1997
), Experimental investigation of the nonlinear response of a hanging cable, Part I: Local analysis
, Nonlinear Dyn.
14
, 89
–117
.91.
Lee
CL
, and Perkins
NC
(1995
), Three-dimensional oscillations of suspended cables involving simultaneous internal resonances
, Nonlinear Dyn.
8
, 45
–63
.92.
Chang
WK
, and Ibrahim
RA
(1997
), Multiple internal resonance in suspended cables under random in-plane loading
, Nonlinear Dyn.
12
, 275
–303
.93.
Benedettini
F
, Rega
G
, and Alaggio
R
(1995
), Nonlinear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions
, J. Sound Vib.
182
, 775
–798
.94.
Ibrahim
RA
, and Chang
WK
(1999
), Stochastic excitation of suspended cables involving three simultaneous internal resonances using Monte Carlo simulation
, Comput. Methods Appl. Mech. Eng.
168
, 285
–304
.95.
Rega
G
, Lacarbonara
W
, Nayfeh
AH
, and Chin
CM
(1999
), Multiple resonances in suspended cables: Direct versus reduced-order models
, Int. J. Non-Linear Mech.
34
, 901
–924
.96.
Nayfeh
AH
, Arafat
HN
, Chin
CM
, and Lacarbonara
W
(2002
), Multimode interactions in suspended cables
, J. Vib. Control
8
, 337
–387
.97.
Benedettini F, and Rega G (1994), Analysis of finite oscillations of elastic cables under internal/external resonance conditions, Nonlinear and Stochastic Dynamics ASME, AMD-192, 39–46.
98.
Irvine
HM
, and Griffin
JH
(1976
), On the dynamic response of a suspended cable
, Earthquake Eng. Struct. Dyn.
4
, 389
–402
.99.
Rega
G
, and Luongo
A
(1980
), Natural vibrations of suspended cables with flexible supports
, Comput. Struct.
12
, 65
–75
.100.
Ni
YQ
, Lou
WJ
, and Ko
JM
(2000
), A hybrid pseudo-force/Laplace transform method for non-linear transient response of a suspended cable
, J. Sound Vib.
238
, 189
–214
.101.
Zheng
G
, Ko
JM
, and Ni
YQ
(2002
), Super-harmonic and internal resonances of a suspended cable with nearly commensurable natural frequencies
, Nonlinear Dyn.
30
, 55
–70
.102.
Pilipchuk
VN
, and Ibrahim
RA
(1997
), Strong nonlinear modal interaction in shallow suspended cables with oscillating ends
, Chaos, Solitons Fractals
8
, 637
–657
.103.
Pilipchuk
VN
, and Ibrahim
RA
(1999
), Nonlinear modal interactions in shallow suspended cables
, J. Sound Vib.
227
, 1
–28
.104.
Cheng
SP
, and Perkins
NC
(1992
), Closed-form vibration analysis of sagged cable/mass suspensions
, ASME J. Appl. Mech.
59
, 923
–928
.105.
Benedettini
F
, and Rega
G
(1997
), Experimental investigation of the nonlinear response of a hanging cable, Part II: Global analysis
, Nonlinear Dyn.
14
, 119
–138
.106.
Molteno
TCA
, and Tufillaro
NB
(1990
), Torus doubling and chaotic string vibrations: Experimental results
, J. Sound Vib.
137
, 327
–330
.107.
O’Reilly
O
(1993
), Global bifurcations in the forced vibration of a damped string
, Int. J. Non-Linear Mech.
28
, 337
–351
.108.
Di Egidio
A
, Luongo
A
, and Vestroni
F
(1996
), Nonstationary nonplanar free motions of an orbiting string with multiple internal resonances
, Meccanica
31
, 363
–381
.109.
Alaggio
R
, and Rega
G
(2000
), Characterizing bifurcations and classes of motion in the transition to chaos through 3D-tori of a continuous experimental system in solid mechanics
, Physica D
137
, 70
–93
.110.
Rega
G
, and Alaggio
R
(2001
), Spatio-temporal dimensionality in the overall complex dynamics of an experimental cable/mass system
, Int. J. Solids Struct.
38
, 2049
–2068
.111.
Takens F (1981), Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, DA Rand and LS Young (eds), Springer Lecture Notes in Mathematics, 898, New York, 266–281.
112.
Holmes P, Lumley JL, and Berkooz G (1996), Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monographs on Mechanics, Cambridge Univ Press, New York.
113.
Ruelle
D
, and Takens
F
(1971
), On the nature of turbulence
, Commun. Math. Phys.
20
, 167
–192
.114.
Newhouse
S
, Ruelle
D
, and Takens
F
(1978
), Occurrence of strange axiom-A attractors near quasiperiodic flows on Tm m⩾3,
Commun. Math. Phys.
64
, 35
–40
.115.
Battelino
PM
, Grebogi
C
, Ott
E
, and Yorke
JA
(1989
), Chaotic attractors on a 3-torus, and the torus break-up
, Physica D
39
, 299
–314
.116.
Baesens
C
, Guckenheimer
J
, Kim
S
, and MacKay
RS
(1991
), Three coupled oscillators: Mode-locking, global bifurcation and toroidal chaos
, Physica D
49
, 387
–475
.117.
Anishchenko
VS
, Safonova
MA
, Feudel
U
, and Kurths
J
(1994
), Bifurcations and transition to chaos through three-dimensional tori
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
4
, 595
–607
.118.
Shilnikov L, Shilnikov A, Turaev D, and O Chua L (1998), Methods of Qualitative Theory in Nonlinear Dynamics, World Scientific Series on Nonlinear Science Series A (4), World Scientific Publ, Singapore.
119.
Glendinning
P
, and Sparrow
C
(1984
), Local and global behaviour near homoclinic orbits
, J. Stat. Phys.
35
, 645
–696
.120.
Rucklidge
AM
(2001
), Global bifurcations in the Takens-Bogdanov normal form with D4 symmetry near the O2 limit
, Phys. Lett. A
284
, 99
–111
.121.
Alaggio R, and Rega G (2002), Homoclinic bifurcation to chaos in a suspended cable-mass: Experimental characterization and theoretical interpretative model, XXX Summer School, Advanced Problems in Mechanics (Book of Abstracts, 19), St Petersburg, Russia.
122.
Rega G, Benedettini F, and Alaggio R (1997b), Experimental and theoretical investigation of nonlinear dynamics in an elastic system with initial curvature, Interaction between Dynamics and Control in Advanced Mechanical Systems, DH van Campen (ed), Solid Mechanics and Its Applications, 52, Kluwer, Dordrecht, 337–344.
123.
Alaggio R, and Rega G (2001), Exploiting results of experimental nonlinear dynamics for reduced-order modeling of a suspended cable, ASME Des. Eng. Tech. Conf., 18th Bienn. Conf. Mechanical Vibration and Noise, DETC01/VIB-21554, CD-Rom, Pittsburgh PA.
124.
Nayfeh AH, and Balachandran B (1995), Applied Nonlinear Dynamics, Wiley, New York.
125.
Rega
G
(1996
), Non-linearity, bifurcation and chaos in the finite dynamics of different cable models
, Chaos, Solitons Fractals
7
, 1507
–1536
.126.
Steindl
A
, and Troger
H
(2001
), Some computational efficient strategies for dimension reduction
, Int. J. Solids Struct.
38
, 2131
–2147
.127.
Rega G, and Troger H (eds) (2005), Dimension Reduction of Dynamical Systems: Methods, Modles, Applications, Special Issue of Nonlinear Dyn. (to appear).
128.
Cusumano
JP
, and Sharkady
MT
(1995
), An experimental study of bifurcation, chaos and dimensionality in a system forced through a bifurcation parameter
, Nonlinear Dyn.
8
, 467
–489
.129.
Kappagantu
RV
, and Feeny
BF
(2000
), Part II: Proper orthogonal modal modeling of a frictionally excited beam
, Nonlinear Dyn.
23
, 1
–11
.130.
Georgiou IT, and Schwartz IB (2001), POD (K-L) analysis of the interaction free dynamics of a beam/pendulum configuration: An experimental study, ASME Des. Eng. Tech. Conf., 18th Bienn. Conf. Mechanical Vibration and Noise, DETC01/VIB-21409, CD-Rom, Pittsburgh PA.
131.
Ma X, Vakakis AF, and Bergman LA (2001), Karhunen-Loeve decomposition for reducing the order and monitoring the dynamics of coupled oscillators, ASME Des. Eng. Tech. Conf., 18th Bienn Conf. Mechanical Vibration and Noise, DETC01/VIB-21404, CD-Rom, Pittsburgh PA.
132.
Mane´ R (1981), On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical Systems and Turbulence, DA Rand and LS Young (eds), Springer Lecture Notes in Mathematics, 898, New York, 230–242.
133.
Cusumano
JP
, and Bai
BY
(1993
), Period-infinity periodic motions, chaos, and spatial coherence in a 10 degree of freedom impact oscillator
, Chaos, Solitons Fractals
3
, 515
–535
.134.
Feeny
BF
, and Kappagantu
R
(1998
), On the physical interpretation of proper orthogonal modes in vibrations
, J. Sound Vib.
211
, 607
–616
.135.
Kerschen
G
, and Golinval
JC
(2002
), Physical interpretation of the proper orthogonal modes using the singular value decomposition
, J. Sound Vib.
249
, 849
–865
.136.
Lin
WZ
, Lee
KH
, Lu
P
, Lim
SP
, and Liang
YC
(2002
), The relationship between eigenfunctions of Karhunen-Loe`ve decomposition and the modes of distributed parameter vibration system
, J. Sound Vib.
252
, 527
–544
.137.
Ma
X
, and Vakakis
AF
(1999
), Karhunen-Loeve decomposition of the transient dynamics of a multibay truss
, AIAA J.
37
, 939
–946
.138.
Kappagantu
R
, and Feeny
BF
(1999
), An optimal modal reduction of a system with frictional excitation
, J. Sound Vib.
224
, 863
–877
.139.
Azeez
MFA
, and Vakakis
AF
(2001
), Proper orthogonal decomposition (POD) of a class of vibroimpact oscillations
, J. Sound Vib.
240
, 859
–889
.140.
Amabili M, Sarkar A, and Pai¨doussis MP (2003), Reduced-order P.O.D. models for nonlinear vibrations of cylindrical shells, ASME Int. Mech. Eng. Congr., IMECE2003-55445, CD-Rom, Washington, DC.
141.
Kerschen
G
, Feeny
BF
, and Golinval
JC
(2003
), On the exploitation of chaos to build reduced-order models
, Comput. Meth. Appl. Mech. Eng.
192
, 1785
–1795
.142.
Georgiou IT (2005), Advanced proper orthogonal decomposition tools: Using reduced order models to identify normal modes of oscillation and slow invariant manifolds in the dynamics of planar nonlinear rods, Nonlinear Dyn. (accepted).
143.
Troger H, and Steindl A (1991), Nonlinear Stability and Bifurcation Theory: An Introduction for Engineers and Applied Scientists, Springer-Verlag, Wien.
144.
Rega G, Srinil N, Lacarbonara W, and Chucheepsakul S (2004), Resonant nonlinear normal modes of inclined sagged cables, Proc. Euromech. Coll. Nonlinear Modes of Vibrating Systems, Fre´jus, Press of ENTPE, Vaulx en Velin, 7–10.
145.
Srinil N (2004), Large-amplitude 3-D dynamic analysis of arbitrarily sagged inclined extensible cables, PhD dissertation, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand.
146.
Feeny
BF
(2002
), On the proper orthogonal modes and normal modes of continuous vibration systems
, J. Vibr. Acoust.
124
, 157
–160
.147.
Kerschen G, Golinval JC, Vakakis AF, and Bergman LA (2005), The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: An overview, Nonlinear Dyn. (accepted).
148.
Sofi
A
, Borino
G
, and Muscolino
G
(2002
), Dynamic analysis of prestressed cables with uncertain Pretension
, Meccanica
37
, 67
–84
.149.
Di Paola M, Muscolino G, and Sofi A (2002), Nonlinear random vibrations of a suspended cable under wind loading, 4th Int. Conf. Comp. Stochastic Mechanics (CSM4), 159–168.
150.
Ibrahim
RA
(2004
), Nonlinear vibrations of suspended cables, Part III: Random excitation and interaction with fluid flow
, Appl. Mech. Rev.
57
, 515
–549
.151.
Lacarbonara W, Paolone A, and Vestroni F (2004), Accurate description of statics and dynamics of nonshallow cables, Tenth Conf. on Nonlinear Vibrations, Stability, and Dynamics of Structures (CDRom of Abstracts), Blacksburg, VA.
152.
Tjavaras
AA
, Zhu
Q
, Liu
Y
, Triantafyllou
MS
, and Yue
DKP
(1998
), The mechanics of highly-extensible cables
, J. Sound Vib.
213
, 709
–737
.153.
Srinil
N
, Rega
G
, and Chucheepsakul
S
(2003
), Large-amplitude three-dimensional free vibrations of inclined sagged elastic cables
, Nonlinear Dyn.
33
, 129
–154
.154.
Srinil N, Rega G, and Chucheepsakul S (2003), Nonlinear interaction in the 3D free vibrations of horizontal and inclined sagged cables, Proc. 5th Int. Symp. Cable Dynamics, Santa Margherita Ligure, AIM, Lie`ge, 77–84.
155.
Gobot
JI
, and Grosenbaugh
MA
(2001
), Application of the generalized-α method to the time integration of the cable dynamic equations
, Comput. Meth. Appl. Mech. Eng.
190
, 4817
–4829
.156.
Newberry
DL
, and Perkins
NC
(1997
), Investigation of resonant tensioning in submerged cables subjected to lateral excitation
, Int. J. Offshore Polar Eng.
7
, 48
–53
.157.
Chang
WK
, Pilipchuk
V
, and Ibrahim
RA
(1997
), Fluid flow-induced nonlinear vibration of suspended cables
, Nonlinear Dyn.
14
, 377
–406
.158.
Hu
HY
, and Jin
DP
(2001
), Nonlinear dynamics of a suspended travelling cable subject to transverse fluid excitation
, J. Sound Vib.
239
, 515
–529
.159.
Kim WJ, and Perkins NC (2001), Coupled slow and fast dynamics of flow excited elastic cable systems, ASME Des. Eng. Tech. Conf., 18th Bienn. Conf. Mechanical Vibration and Noise, DETC 01/VIB-21589, CD-ROM, Pittsburgh PA.
160.
Tanaka H (2003), Aerodynamics of cables, Proc 5th Int Symp Cable Dynamics, Santa Margherita Ligure, AIM, Lie`ge, 11–25.
161.
Lu
CL
, and Perkins
NC
(1994
), Nonlinear spatial equilibria and stability of cables under uni-axial torque and thrust
, ASME J. Appl. Mech.
61
, 879
–886
.162.
Lu
CL
, and Perkins
NC
(1995
), Complex spatial equilibria of U-joint supported cables under torque, thrust and self-weight
, Int. J. Non-Linear Mech.
30
, 271
–285
.163.
Gottlieb
O
, and Perkins
, NC
(1999
), Local and global bifurcation analyses of a spatial cable elastica
, ASME J. Appl. Mech.
66
, 352
–360
.164.
Koh
CG
, Zhang
Y
, and Quek
ST
(1999
), Low-tension cable dynamics: Numerical and experimental studies
, J. Eng. Mech. Div.
125
, 347
–354
.165.
Wu
Q
, Takahashi
K
, and Nakamura
S
(2003
), Non-linear vibrations of cables considering loosening
, J. Sound Vib.
261
, 385
–402
.166.
Wu
Q
, Takahashi
K
, and Nakamura
S
(2004
), Non-linear response of cables subjected to periodic support excitation considering cable loosening
, J. Sound Vib.
271
, 453
–463
.167.
Luongo A, and Piccardo G (1996), On the influence of the torsional stiffness on non-linear galloping of suspended cables, Proc. 2nd Eur. Nonlinear Oscillation Conf., Prague, 273–276.
168.
Goyal S, Perkins NC, and Lee CL (2003), Torsional buckling and writhing dynamics of elastic cables and DNA, ASME Des. Eng. Techn. Conf., 19th Bienn. Conf. Mechanical Vibration and Noise, DETC2003/MECH-48322, CD-Rom, Chicago IL.
169.
Koh
CG
, and Rong
Y
(2004
), Dynamic analysis of large displacement cable motion with experimental verification
, J. Sound Vib.
272
, 187
–206
.170.
Ni
YQ
, Ko
JM
, and Zheng
G
(2002
), Dynamic analysis of large-diameter sagged cables taking into account flexural rigidity
, J. Sound Vib.
257
, 301
–319
.171.
Triantafyllou
MS
, and Yue
DKP
(1995
), Damping amplification in highly extensible hysteretic cables
, J. Sound Vib.
186
, 355
–368
.172.
Leamy
MJ
, and Gottlieb
O
(2000
), Internal resonances in whirling strings involving longitudinal dynamics and material non-linearities
, J. Sound Vib.
236
, 683
–703
.173.
Han
SM
, and Grosenbaugh
MA
(2004
), Non-linear free vibration of a cable against a straight obstacle
, J. Sound Vib.
273
, 337
–361
.174.
Chen LQ, (2005) Analysis and control of transverse vibrations of axially moving strings, Appl. Mech. Rev. (to appear).
175.
Perkins
NC
, and Mote
Jr CD
(1987
), Three-dimensional vibration of travelling elastic cables
, J. Sound Vib.
114
, 325
–340
.176.
Luo
ACJ
, and Mote
Jr CD
(2000
), Equilibrium solutions and existence for traveling, arbitrarily sagged elastic cables
, ASME J. Appl. Mech.
67
, 148
–154
.177.
Wang
Y
, and Luo
ACJ
(2004
), Dynamics of traveling, inextensible cables
, Comm. Nonlin. Sci. Num. Simul.
9
, 531
–542
.178.
Wang
YM
(2000
), The transient dynamics of a cable-mass system due to the motion of an attached accelerating mass
, Int. J. Solids Struct.
37
, 1361
–1383
.179.
Al-Qassab
M
, Nair
S
, and O’Leary
J
(2003
), Dynamics of an elastic cable carrying a moving mass particle
, Nonlinear Dyn.
33
, 11
–32
.180.
Muscolino G, and Sofi A (2003), Dynamics of a suspended cable under moving masses, Proc. 5th Int. Symp. Cable Dynamics, Santa Margherita Ligure, AIM, Lie`ge, 109–116.
181.
Fujino
Y
, Warnitchai
P
, and Pacheco
BM
(1993
), Active stiffness control of cable vibration
, ASME J. Appl. Mech.
60
, 953
–984
.182.
Fujino
Y
, and Susumpow
T
(1994
), An experimental study on active control of in-plane cable vibration by axial support motion
, Earthquake Eng. Struct. Dyn.
23
, 1283
–1297
.183.
Canbolat
H
, Dawson
D
, Rahn
CD
, and Nagarkatti
S
(1998
), Adaptive boundary control of out-of-plane cable vibration
, ASME J. Appl. Mech
65
, 963
–969
.184.
Gehle
RW
, and Masri
SF
(1998
), Active control of shallow, slack cable using the parametric control of end tension
, Nonlinear Dyn.
17
, 77
–94
.185.
Gattulli
V
, and Vestroni
F
(2000
), Nonlinear strategies for longitudinal control in the stabilization of an oscillating suspended cable
, Dyn. Control
10
, 359
–374
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