This review article starts by addressing the mathematical principles of the perturbation method of multiple scales in the context of mechanical systems which are defined by weakly nonlinear ordinary differential equations. At this stage the paper investigates some different forms of typical nonlinearities which are frequently encountered in machine and structural dynamics. This leads to conclusions relating to the relevance and scope of this popular and versatile method, its strengths, its adaptability and potential for different variant forms, and also its weaknesses. Key examples from the literature are used to develop and consolidate these themes. In addition to this the paper examines the role of term-ordering, the integration of the so-called small (ie, perturbation) parameter within system constants, nondimensionalization and time-scaling, series truncation, inclusion and exclusion of higher order nonlinearities, and typical problems in the handling of secular terms. This general discussion is then applied to models of the dynamics of space tethers given that these systems are nonlinear and necessarily highly susceptible to modelling accuracy, thus offering a rigorous and testing applications case-study area for the multiple scales method. The paper concludes with comments on the use of variants of the multiple scales method, and also on the constraints that the method can bring to expectations of modelling accuracy. This review article contains 134 references.
Issue Section:
Review Articles
1.
Nayfeh AH and Mook DT (1979), Nonlinear Oscillations, Wiley Interscience, New York.
2.
Nayfeh AH (1973), Perturbation Methods, John Wiley, New York.
3.
Jordan DW and Smith P (1977), Nonlinear Ordinary Differential Equations, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, Oxford, UK.
4.
Cartmell MP (1990), Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman and Hall.
5.
Thomsen JJ (1997), Vibrations and Stability, Order and Chaos, McGraw-Hill, Maidenhead, UK.
6.
Warmin´ski JA (2001), Drgania Regularne I Chaotyczne, Politechnika Lubelska, Lublin, Poland (in Polish).
7.
Sanders JA and Verhulst F (1985), Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59, Springer Verlag, New York.
8.
Khanin
R
and Cartmell
MP
(1999), Applying the perturbation method of multiple scales, Math. Educ. Res. 8(2), 19–26.9.
Khanin
R
, Cartmell
MP
, and Gilbert
A
(2000
), A computerized implementation of the multiple scales perturbation method using Mathematica
, Comput. Struct.
76
(5
), 565
–575
.10.
Khanin
R
and Cartmell
MP
(2001
), Parallelization of perturbation analysis: Application to large-scale engineering problems
, J. Symbol. Comput.
31
(4
), 461
–473
.11.
Nayfeh
AH
, Chin
C-M
, and Pratt
J
(1997
), Perturbation methods in nonlinear dynamics-application to machining dynamics
, ASME J. Manuf. Sci. Eng.
119
, 485
–493
.12.
Forehand
DIM
and Cartmell
MP
(2001
), On the derivation of the equations of motion for a parametrically excited cantilever beam
, J. Sound Vib.
245
(1
), 165
–177
.13.
Murdock JA (1991), Perturbations: Theory and Methods, John Wiley, New York.
14.
Watt
D
and Cartmell
MP
(1994
), An externally loaded parametric oscillator
, J. Sound Vib.
170
(3
), 339
–364
.15.
Lee
CL
and Lee
C-T
(1997
), A higher order method of multiple scales
, J. Sound Vib.
202
(2
), 284
–287
.16.
Rahman
Z
and Burton
TD
(1989
), On higher-order methods of multiple scales in non-linear oscillations-periodic steady-state response
, J. Sound Vib.
133
(3
), 369
–379
.17.
Rahman
Z
and Burton
TD
(1986
), Large amplitude primary and superharmonic resonances in the duffing oscillator
, J. Sound Vib.
110
(3
), 363
–380
.18.
Luongo
A
, Rega
G
, and Vestroni
F
(1986
), On nonlinear dynamics of planar shear indeformable beams
, ASME J. Appl. Mech.
53
(3
), 619
–624
.19.
Luongo
A
and Paolone
A
(1999
), On the reconstitution problem in the multiple time scale method
, Nonlinear Dyn.
19
, 133
–156
.1.
Hassan
A
(1994
), Use of transformations with the higher order method of multiple scales to determine the steady state periodic response of harmonically excited nonlinear oscillators, Part I: Transformation of derivative
, J. Sound Vib.
178
, 21
–40
; 2.
Part II: Transformation of detuning, J. Sound Vib. 178, 1–19.
1.
Lee
WK
, Yeo
MH
, and Bae
SS
(1997
), Validity of the multiple-scale solution for a subharmonic resonance response of a bar with a nonlinear boundary condition
, J. Sound Vib.
208
(4
), 567
–574
.2.
Bux
SL
and Roberts
JW
(1986
), Nonlinear vibratory interactions in systems of coupled beams
, J. Sound Vib.
104
(3
), 497
–520
.3.
Cartmell
MP
and Roberts
JW
(1988
), Simultaneous combination resonances in an autoparametrically resonant system
, J. Sound Vib.
123
(1
), 81
–101
.4.
Ji
JC
, Yu
L
, and Chen
YS
(1999
), Bifurcation and amplitude modulated motions in a parametrically excited two-degree-of-freedom nonlinear system
, J. Sound Vib.
228
(5
), 1125
–1144
.5.
Nayfeh
AH
, Balachandran
B
, Colbert
MA
, and Nayfeh
MA
(1989
), An experimental investigation of complicated responses of a 2-dof structure
, ASME J. Appl. Mech.
56
, 960
–967
.6.
Anderson
TJ
, Balachandran
B
, and Nayfeh
AH
(1994
), Nonlinear resonances in a flexible cantilever beam
, ASME J. Vibr. Acoust.
116
, 480
–484
.7.
Anderson
TJ
, Nayfeh
AH
, and Balachandran
B
(1996
), Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam
, ASME J. Vibr. Acoust.
118
, 21
–27
.8.
Ibrahim
RA
, Afaneh
AA
, and Lee
BH
(1993
), Structural modal multifurcation with internal resonance, Part 1: Deterministic approach
, ASME J. Vibr. Acoust.
115
(2
), 182
–192
.9.
Dwivedy
SK
and Kar
RC
(1999
), Nonlinear response of a parametrically excited system using higher order method of multiple scales
, Nonlinear Dyn.
20
(2
), 115
–130
.10.
Al-Qaisia
AA
and Hamdan
MN
(1999
), On the steady state response of oscillators with static and inertia nonlinearities
, J. Sound Vib.
223
(1
), 49
–71
.11.
O¨z
HR
, Pakdemirli
M
, O¨zkaya
E
, and Yilmaz
M
(1998
), Nonlinear vibrations of a slightly curved beam resting on a nonlinear elastic foundation
, J. Sound Vib.
212
(2
), 295
–309
.12.
Cartmell MP (1984), Combination instabilities and nonlinear vibratory interactions in beam systems, PhD Thesis, Univ of Edinburgh.
13.
Nayfeh
AH
and Lacarbonara
W
(1997
), On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities
, Nonlinear Dyn.
13
(3
), 203
–220
.14.
Lacarbonara
W
(1999
), Direct treatment and discretizations of nonlinear spatially continuous systems
, J. Sound Vib.
221
(5
), 849
–866
.15.
Nayfeh
AH
, Lacarbonara
W
, and Chin
CM
(1999
), Nonlinear normal modes of buckled beams: Three-to-one and one-to-one internal resonances
, Nonlinear Dyn.
18
(3
), 253
–273
.16.
Lacarbonara
W
, Nayfeh
AH
, and Kreider
W
(1998
), Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: Analysis of a buckled beam
, Nonlinear Dyn.
17
(2
), 95
–117
.17.
Krauss
RW
and Nayfeh
AH
(1999
), Experimental nonlinear identification of a single mode of a transversely excited beam
, Nonlinear Dyn.
18
(1
), 69
–87
.18.
Abe
A
, Kobayashi
Y
, and Yamada
G
(1998
), Analysis of subharmonic resonance of moderately thick antisymmetric angle-ply laminated plates by using method of multiple scales
, J. Sound Vib.
217
(3
), 467
–484
.19.
Abe
A
, Kobayashi
Y
, and Yamada
G
(1998
), Two-mode response of simply supported, rectangular laminated plates
, Int. J. Non-Linear Mech.
33
(4
), 675
–690
.20.
Abe
A
, Kobayashi
Y
, and Yamada
G
(1998
), Three-mode response of simply supported, rectangular laminated plates
, JSME Int. J., Ser. C
41
(1
), 51
–59
.21.
Sorokin
SV
(2000
), Nonlinear oscillations of a baffled elastic plate in heavy fluid loading conditions
, J. Sound Vib.
232
(3
), 619
–643
.22.
Wang
YX
(1998
), Multifrequency resonances of flexible linkages
, Mech. Mach. Theory
33
(3
), 255
–271
.23.
Wang
YX
(1997
), Dynamics of an elastic four bar linkage mechanism with geometric nonlinearities
, Nonlinear Dyn.
14
(4
), 357
–375
.24.
El-Nagar
AM
and El-Bassiouny
AF
(1992
), Response of self-excited 3-degree-of-freedom systems to multifrequency excitations
, Int. J. Theor. Phys.
31
(8
), 1531
–1548
.25.
El-Bassiouny
AF
(1999
), Response of a three-degree-of-freedom system with cubic nonlinearities to harmonic excitation
, Appl. Math. Comput.
104
, 65
–84
.26.
Mazzilli
CEN
and Brasil
RMLRF
(1995
), Effect of static loading on the nonlinear vibrations of a 3-time redundant portal frame—analytical and numerical studies
, Nonlinear Dyn.
8
(3
), 347
–366
.27.
Jin
DP
and Hu
HY
(1998
), Ice-induced nonlinear vibration of an offshore platform
, J. Sound Vib.
214
(3
), 431
–442
.28.
Thomsen
JJ
, (1992
), Chaotic vibrations of non-shallow arches
, J. Sound Vib.
153
(2
), 239
–258
.29.
Nayfeh
AH
and Lacarbonara
W
(1998
), On the discretization of spatially continuous systems with quadratic and cubic nonlinearities
, JSME Int. J., Ser. C
41
(3
), 510
–531
.30.
Jensen
JS
(1997
), Fluid transport due to nonlinear fluid-structure interaction
, J. Fluids Struct.
11
, 327
–344
.31.
Vakakis
A
, Nayfeh
T
, and King
M
(1993
), A multiple scales analysis of nonlinear, localized modes in a cyclic periodic system
, ASME J. Appl. Mech.
60
(2
), 388
–397
.32.
King
ME
and Layne
PA
(1998
), Dynamics of nonlinear cyclic systems with structural irregularity
, Nonlinear Dyn.
15
(3
), 225
–244
.33.
Zhang
L
and Zu
JW
(1998
), Nonlinear vibrations of viscoelastic moving belts, Part I: Free vibration analysis
, J. Sound Vib.
216
(1
), 75
–91
.34.
Ji
Z
and Zu
JW
(1998
), Method of multiple scales for vibration analysis of rotor-shaft systems with nonlinear bearing pedestal model
, J. Sound Vib.
218
(2
), 293
–305
.35.
Young
TH
(1992
), Nonlinear transverse vibrations and stability of spinning disks with nonconstant spinning rate
, ASME J. Vibr. Acoust.
114
(4
), 506
–513
.36.
Lamb
H
and Southwell
RV
(1921
), The vibrations of a spinning disk
, Proc. R. Soc. London, Ser. A
99
, 272
–280
.37.
Chan
SN
, Mottershead
JE
, and Cartmell
MP
(1994
), Parametric resonances at subcritical speeds in disks with rotating frictional loads
, Proc. Inst. Mech. Eng., Part C: Mech. Eng. Sci.
208
(6
), 417
–425
.38.
Mottershead
JE
and Chan
SN
(1995
), Flutter instability of circular disks with frictional follower loads
, ASME J. Vibr. Acoust.
117
(1
), 161
–163
.39.
Asfar
KR
(1992
), Effect of nonlinearities in elastomeric material dampers on torsional vibration control
, Int. J. Non-Linear Mech.
27
(6
), 947
–954
.40.
Oueini
SS
, Chin
C-M
, and Nayfeh
AH
(1995
), Dynamics of a cubic nonlinear vibration absorber
, Nonlinear Dyn.
20
(3
), 283
–295
.41.
Yang
S
, Nayfeh
AH
, and Mook
DT
(1998
), Combination resonances in the response of the duffing oscillator to a three-frequency excitation
, Acta Mech.
131
(3–4
), 235
–245
.42.
Hu
HY
, Dowell
EH
, and Virgin
LN
(1998
), Resonances of a harmonically forced duffing oscillator with time delay state feedback
, Nonlinear Dyn.
15
(4
), 311
–327
.43.
Virgin
LN
and Cartee
LA
(1991
), A note on the escape from a potential well
, Int. J. Non-Linear Mech.
26
(3–4
), 449
–452
.44.
Nayfeh
AH
(1984
), Interaction of fundamental parametric resonances with subharmonic resonances of order one-half
, J. Sound Vib.
96
(3
), 333
–340
.45.
Cartmell
MP
and Roberts
JW
(1987
), Simultaneous combination resonances in a parametrically excited cantilever beam
, Strain J. Brit. Soc. Strain Measurement
23
(3
), 117
–126
.46.
Nayfeh
AH
and Arafat
HN
(1998
), Nonlinear response of cantilever beams to combination and subcombination resonances
, Shock Vib.
5
(5–6
), 277
–288
.47.
Oueini
SS
and Nayfeh
AH
(1999
), Single-mode control of a cantilever beam under principal parametric excitation
, J. Sound Vib.
224
(1
), 33
–47
.48.
Huang
Y-M
and Lee
C-Y
(1998
), Dynamics of a rotating rayleigh beam subject to a repetitively travelling force
, Int. J. Mech. Sci.
40
(8
), 779
–792
.49.
Nayfeh
AH
(1998
), Reduced-order models of weakly nonlinear spatially continuous systems
, Nonlinear Dyn.
16
(2
), 105
–125
.50.
Lee
WK
and Soh
KY
(1994
), Nonlinear analysis of the forced response of a beam with 3 mode interaction
, Nonlinear Dyn.
6
(1
), 49
–68
.51.
Chin
C-M
and Nayfeh
AH
(1997
), Three-to-one internal resonances in hinged-clamped beams
, Nonlinear Dyn.
12
(2
), 129
–154
.52.
Haddow
AG
, Barr
ADS
, and Mook
DT
(1984
), Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure
, J. Sound Vib.
97
(3
), 451
–473
.53.
Nayfeh
TA
, Nayfeh
AH
, and Mook
DT
(1994
), A theoretical and experimental investigation of a three degree-of-freedom structure
, Nonlinear Dyn.
6
(3
), 353
–374
.54.
Haxton
RS
and Barr
ADS
(1972
), The autoparametric vibration absorber
, ASME J. Eng. Ind.
94
, 119
–125
.55.
Cartmell
MP
and Lawson
J
(1994
), Performance enhancement of an autoparametric vibration absorber by means of computer control
, J. Sound Vib.
177
(2
), 173
–195
.56.
Hatwal
H
, Mallik
AK
, and Ghosh
A
(1982
), Nonlinear vibrations of a harmonically excited autoparametric system
, J. Sound Vib.
81
(2
), 153
–164
.57.
Tondl A, Ruijgrok T, Verhulst F, and Nabergoj R (2000), Autoparametric Resonance in Mechanical Systems, Cambridge Univ Press, Cambridge, UK.
58.
Nayfeh
AH
and Asfar
KR
(1986
), Response of a bar constrained by a nonlinear spring to a harmonic excitation
, J. Sound Vib.
105
(1
), 1
–15
.59.
Lee
WK
and Park
HD
(1999
), Second order approximation for chaotic responses of a harmonically excited spring-pendulum system
, Int. J. Non-Linear Mech.
34
(4
), 749
–757
.60.
Lee
WK
and Park
HD
(1997
), Chaotic dynamics of a harmonically excited spring-pendulum system with internal resonance
, Nonlinear Dyn.
14
(3
), 211
–229
.61.
Balachandran
B
and Khan
KA
(1996
), Spectral analysis of nonlinear interactions
, Mech. Syst. Signal Process.
10
(6
), 711
–727
.62.
Kondou
T
and Yagasaki
K
(1995
), Some recent topics on nonlinear vibration and chaos
, Trans. Jpn. Soc. Mech. Eng., Ser. C
61
, 746
–751
.63.
Iwan
WD
and Stahl
KJ
(1973
), The response of an elastic disk with a moving mass system
, ASME J. Appl. Mech.
40
, 445
–451
.64.
Mote
CD
(1970
), Stability of circular plates subjected to moving loads
, J. Franklin Inst.
290
, 329
–344
.65.
Shen
IY
(1993
), Response of a stationary, damped, circular plate under a rotating slider bearing system
, ASME J. Vibr. Acoust.
115
(1
), 65
–69
.66.
Shen
IY
and Mote
CD
(1991
), On the mechanisms of instability of a circular plate under a rotating spring-mass-dashpot system
, J. Sound Vib.
148
(2
), 307
–318
.67.
Ouyang
H
, Mottershead
JE
, Cartmell
MP
, and Friswell
MI
(1997
), Parametric resonances in an annular disk, with a rotating system of distributed mass and elasticity; and the effects of friction and damping
, Proc. R. Soc. London, Ser. A
453
, 1
–19
.68.
Cartmell MP and Ouyang H (2000), Dynamic instabilities in spinning disks, Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, M Wiercigroch and B de Kraker (eds), World Scientific, 313–341.
69.
Wiercigroch M and de Kraker B (eds), (2000), Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, World Scientific, 444.
70.
Sueoka
A
, Ryu
T
, Fujiyama
M
, and Yoshitake
Y
(1995
), Quenching of frictional vibration of a rotating circular plate by dynamic absorbers
, JSME Int. J., Ser. C
38
(3
), 441
–449
.71.
Heuer
R
(1994
), Large flexural vibrations of thermally stressed layered shallow shells
, Nonlinear Dyn.
5
(1
), 25
–38
.72.
Rega
G
, Lacarbonara
W
, Nayfeh
AH
, and Chin
C-M
(1999
), Multiple resonances in suspended cables: Direct versus reduced-order models
, Int. J. Non-Linear Mech.
34
(5
), 901
–924
.73.
Merchant
ME
(1944
), Basic mechanics in the metal cutting process
, Trans. ASME
11
, 168
–175
.74.
Grabec
I
(1986
), Chaos generated by the cutting process
, Phys. Lett. A
117
(8
), 384
–386
.75.
Wiercigroch
M
(1997
), Chaotic vibration of a simple model of the machine tool cutting process system
, ASME J. Vibr. Acoust.
119
(3
), 468
–475
.76.
Warmin´ski JA, Litak G, Lipski J, Wiercigroch M, and Cartmell MP (2000), Chaotic vibrations in the regenerative cutting process, IUTAM/IFToMM Symp on Synthesis of Nonlinear Dynamical Systems, E Lavendis and M Zakrzhevsky (eds), Kluwer Academic Publishers, Dordrecht, Netherlands, 275–284.
77.
Warmin´ski
JA
, Cartmell
MP
, Khanin
R
, Wiercigroch
M
, and Litak
G
(2001
), Approximate analytical solutions for primary chatter in the nonlinear metal cutting model
, J. Sound Vib.
259
(4
), 917
–933
.78.
Ganesan
R
(1996
), Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances
, Mech. Mach. Theory
31
(6
), 781
–798
.79.
Moon
J
and Wickert
JA
(1997
), Nonlinear vibration of power transmission belts
, J. Sound Vib.
200
(4
), 419
–431
.80.
Bogoliubov NN and Mitropolski YA (1961), Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.
81.
Zhang
L
and Zu
JW
(1998
), Nonlinear vibrations of viscoelastic moving belts, Part 2: forced vibration analysis
, J. Sound Vib.
216
(1
), 93
–105
.82.
Love AEH (1944), A Treatise on the Mathematical Theory of Elasticity, 4th Edition, 1st American Press, New York.
83.
Yabuno
H
, Ide
Y
, and Aoshima
N
(1998
), Nonlinear analysis of a parametrically excited cantilever beam
, JSME Int. J., Ser. C
41
(3
), 555
–562
.84.
Guckenheimer J and Holmes P (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
85.
Kar
RC
and Sujata
T
(1990
), Parametric instability of an elastically restrained cantilever beam
, Comput. Struct.
34
(3
), 469
–475
.86.
Kar
RC
and Sujata
T
(1992
), Dynamic stability of a rotating, pretwisted and preconed cantilever beam including coriolis effects
, Comput. Struct.
42
(2
), 741
–750
.87.
Kim
J-H
and Choo
Y-S
(1998
), Dynamic stability of a free-free Timoshenko beam subjected to a pulsating follower force
, J. Sound Vib.
216
(4
), 623
–636
.88.
Chin
C-M
and Nayfeh
AH
(1999
), Three-to-one internal resonances in parametrically excited hinged-clamped beams
, Nonlinear Dyn.
20
(2
), 131
–158
.89.
Cederbaum
G
(1992
), Analysis of parametrically excited laminated shells
, Int. J. Mech. Sci.
34
(3
), 241
–250
.90.
Young
TH
and Chen
FY
(1993
), Stability of fluttered panels subjected to in-plane harmonic forces
, AIAA J.
31
(9
), 1667
–1673
.91.
Hansen
MH
(2000
), Effect of high frequency excitation on natural frequencies of spinning disks
, J. Sound Vib.
234
(4
), 577
–589
.92.
Young
TH
and Liou
GT
(1993
), Dynamic response of rotor-bearing systems with time-dependent spin rates
, ASME J. Eng. Gas Turbines Power
115
(2
), 239
–245
.93.
Meirovitch L (1970), Methods of Analytical Dynamics, McGraw-Hill, New York.
94.
El-Sayad
MA
, Hanna
SN
, and Ibrahim
RA
(1999
), Parametric excitation of nonlinear elastic systems involving hydrodynamic sloshing impact
, Nonlinear Dyn.
18
(1
), 25
–50
.95.
Belhaq
M
and Houssni
M
(1999
), Quasi-periodic oscillations, chaos and suppression of chaos in a nonlinear oscillator driven by parametric and external excitations
, Nonlinear Dyn.
18
(1
), 1
–24
.96.
Asfar
KR
and Masoud
KK
(1994
), Damping of parametrically excited single-degree-of-freedom systems
, Int. J. Non-Linear Mech.
29
(3
), 421
–428
.97.
Nayfeh
AH
(1983
), Response of multi-degree-of-freedom systems with quadratic nonlinearities to a harmonic parametric resonance
, J. Sound Vib.
90
(2
), 237
–244
.98.
Chin
C
and Nayfeh
AH
(1995
), Parametrically excited nonlinear two-degree-of-freedom systems with nonsemisimple one-to-one resonance
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
5
(3
), 725
–740
.99.
Tcherniak
D
(1999
), The influence of fast excitation on a continuous system
, J. Sound Vib.
227
(2
), 343
–360
.100.
Fidlin
A
(2000
), On asymptotic properties of systems with strong and very strong high frequency excitation
, J. Sound Vib.
235
(2
), 219
–233
.101.
Thomsen
JJ
(1995
), Chaotic dynamics of the partially follower-loaded elastic double pendulum
, J. Sound Vib.
188
(3
), 385
–405
.102.
Smith
P
(1998
), The multiple scales method, homoclinic bifurcation and Melnikov’s method for autonomous systems
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
8
(11
), 2099
–2105
.103.
Chin
C
, Nayfeh
AH
, and Mook
DT
(1995
), The response of a nonlinear system with a nonsemisimple one-to-one resonance to a combination parametric resonance
, Int. J. Bifurcation Chaos Appl. Sci. Eng.
5
(4
), 971
–982
.104.
Moremedi
GM
, Mason
DP
, and Gorringe
VM
(1993
), On the limit cycle of a generalized van der pol equation
, Int. J. Non-Linear Mech.
28
(2
), 237
–250
.105.
Rong
H
, Xu
W
, and Fang
T
(1998
), Principal response of duffing oscillator to combined deterministic and narrow-band random parametric excitation
, J. Sound Vib.
210
(4
), 483
–515
.106.
Cartmell MP (1998), Generating velocity increments by means of a spinning motorized tether, Paper AIAA 98-9739, 34th AIAA/ASME/SAE/ASEE Joint Propulsion Conf and Exhibit, July 13–15, Cleveland OH.
107.
Cartmell MP and Ziegler SW (1999), Symmetrically laden motorized tethers for two-way interplanetary payload exchange, Paper AIAA 99-2840, 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conf and Exhibit, June 20–24, Los Angeles CA.
108.
Cartmell MP and Ziegler SW (2000), Terrestrial scale model testing of a motorized propulsion tether, Paper AIAA 2000-3612, 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conf and Exhibit, July 16–19 Huntsville AL.
109.
Cartmell MP and Ziegler SW (2001), Experimental scale model testing of a motorized momentum exchange propulsion tether, Paper AIAA 2001-3914, 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conf and Exhibit, July 8–11, Salt Lake City UT.
110.
Ziegler SW and Cartmell MP (2000), Investigating the use of motorized tethers for payload orbital transfer, Paper AIAA 2000-4529, AIAA/AAS Astrodynamics Specialist Conf and Exhibit, Aug 14–17, Denver CO.
111.
Ziegler SW and Cartmell MP (2001), On the validity of recent pre-dictions for tethers on elliptical orbits, Paper AAS 01-233, 11th AAS/AIAA Space Flight Mechanics Meeting, Feb 11–15, Santa Barbara CA.
112.
Ziegler
SW
and Cartmell
MP
(2001
), Using motorized tethers for payload orbital transfer
, J. Spacecr. Rockets
38
(6
), 904
–913
.113.
Ziegler SW (2003), Rigid body dynamics of tethers in space, PhD Thesis, Univ of Glasgow, Scotland, UK.
114.
Chobotov VA (1996), Orbital mechanics, Orbital Mechanics, 2nd Edition, AIAA, Reston VA.
Copyright © 2003
by ASME
You do not currently have access to this content.
