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REVIEW ARTICLES

Multiple scales analyses of the dynamics of weakly nonlinear mechanical systems

[+] Author and Article Information
MP Cartmell, SW Ziegler, R Khanin, DIM Forehand

Department of Mechanical Engineering, University of Glasgow, G12 8QQ, Scotland, UKDepartment of Mechanical, Aerospace and Manufacturing Engineering, UMIST, Sackville Street, PO Box 88, Manchester M60 1QD, UK

Appl. Mech. Rev 56(5), 455-492 (Aug 29, 2003) (38 pages) doi:10.1115/1.1581884 History: Online August 29, 2003
Copyright © 2003 by ASME
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References

Nayfeh AH and Mook DT (1979), Nonlinear Oscillations, Wiley Interscience, New York.
Nayfeh AH (1973), Perturbation Methods, John Wiley, New York.
Jordan DW and Smith P (1977), Nonlinear Ordinary Differential Equations, Oxford Applied Mathematics and Computing Science Series, Oxford University Press, Oxford, UK.
Cartmell MP (1990), Introduction to Linear, Parametric and Nonlinear Vibrations, Chapman and Hall.
Thomsen JJ (1997), Vibrations and Stability, Order and Chaos, McGraw-Hill, Maidenhead, UK.
Warmiński JA (2001), Drgania Regularne I Chaotyczne, Politechnika Lubelska, Lublin, Poland (in Polish).
Sanders JA and Verhulst F (1985), Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, 59 , Springer Verlag, New York.
Khanin  R and Cartmell  MP (1999), Applying the perturbation method of multiple scales, Math. Educ. Res. 8 (2), 19–26.
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.
Khanin  R and Cartmell  MP (2001), Parallelization of perturbation analysis: Application to large-scale engineering problems, J. Symbol. Comput. 31(4), 461–473.
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.
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.
Murdock JA (1991), Perturbations: Theory and Methods, John Wiley, New York.
Watt  D and Cartmell  MP (1994), An externally loaded parametric oscillator, J. Sound Vib. 170(3), 339–364.
Lee  CL and Lee  C-T (1997), A higher order method of multiple scales, J. Sound Vib. 202(2), 284–287.
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.
Rahman  Z and Burton  TD (1986), Large amplitude primary and superharmonic resonances in the duffing oscillator, J. Sound Vib. 110(3), 363–380.
Luongo  A, Rega  G, and Vestroni  F (1986), On nonlinear dynamics of planar shear indeformable beams, ASME J. Appl. Mech. 53(3), 619–624.
Luongo  A and Paolone  A (1999), On the reconstitution problem in the multiple time scale method, Nonlinear Dyn. 19, 133–156.
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; Part II: Transformation of detuning, J. Sound Vib. 178 , 1–19.
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.
Bux  SL and Roberts  JW (1986), Nonlinear vibratory interactions in systems of coupled beams, J. Sound Vib. 104(3), 497–520.
Cartmell  MP and Roberts  JW (1988), Simultaneous combination resonances in an autoparametrically resonant system, J. Sound Vib. 123(1), 81–101.
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.
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.
Anderson  TJ, Balachandran  B, and Nayfeh  AH (1994), Nonlinear resonances in a flexible cantilever beam, ASME J. Vibr. Acoust. 116, 480–484.
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.
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.
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.
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.
Öz  HR, Pakdemirli  M, Özkaya  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.
Cartmell MP (1984), Combination instabilities and nonlinear vibratory interactions in beam systems, PhD Thesis, Univ of Edinburgh.
Nayfeh  AH and Lacarbonara  W (1997), On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities, Nonlinear Dyn. 13(3), 203–220.
Lacarbonara  W (1999), Direct treatment and discretizations of nonlinear spatially continuous systems, J. Sound Vib. 221(5), 849–866.
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.
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.
Krauss  RW and Nayfeh  AH (1999), Experimental nonlinear identification of a single mode of a transversely excited beam, Nonlinear Dyn. 18(1), 69–87.
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.
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.
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.
Sorokin  SV (2000), Nonlinear oscillations of a baffled elastic plate in heavy fluid loading conditions, J. Sound Vib. 232(3), 619–643.
Wang  YX (1998), Multifrequency resonances of flexible linkages, Mech. Mach. Theory 33(3), 255–271.
Wang  YX (1997), Dynamics of an elastic four bar linkage mechanism with geometric nonlinearities, Nonlinear Dyn. 14(4), 357–375.
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.
El-Bassiouny  AF (1999), Response of a three-degree-of-freedom system with cubic nonlinearities to harmonic excitation, Appl. Math. Comput. 104, 65–84.
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.
Jin  DP and Hu  HY (1998), Ice-induced nonlinear vibration of an offshore platform, J. Sound Vib. 214(3), 431–442.
Thomsen  JJ, (1992), Chaotic vibrations of non-shallow arches, J. Sound Vib. 153(2), 239–258.
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.
Jensen  JS (1997), Fluid transport due to nonlinear fluid-structure interaction, J. Fluids Struct. 11, 327–344.
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.
King  ME and Layne  PA (1998), Dynamics of nonlinear cyclic systems with structural irregularity, Nonlinear Dyn. 15(3), 225–244.
Zhang  L and Zu  JW (1998), Nonlinear vibrations of viscoelastic moving belts, Part I: Free vibration analysis, J. Sound Vib. 216(1), 75–91.
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.
Young  TH (1992), Nonlinear transverse vibrations and stability of spinning disks with nonconstant spinning rate, ASME J. Vibr. Acoust. 114(4), 506–513.
Lamb  H and Southwell  RV (1921), The vibrations of a spinning disk, Proc. R. Soc. London, Ser. A 99, 272–280.
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.
Mottershead  JE and Chan  SN (1995), Flutter instability of circular disks with frictional follower loads, ASME J. Vibr. Acoust. 117(1), 161–163.
Asfar  KR (1992), Effect of nonlinearities in elastomeric material dampers on torsional vibration control, Int. J. Non-Linear Mech. 27(6), 947–954.
Oueini  SS, Chin  C-M, and Nayfeh  AH (1995), Dynamics of a cubic nonlinear vibration absorber, Nonlinear Dyn. 20(3), 283–295.
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.
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.
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.
Nayfeh  AH (1984), Interaction of fundamental parametric resonances with subharmonic resonances of order one-half, J. Sound Vib. 96(3), 333–340.
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.
Nayfeh  AH and Arafat  HN (1998), Nonlinear response of cantilever beams to combination and subcombination resonances, Shock Vib. 5(5–6), 277–288.
Oueini  SS and Nayfeh  AH (1999), Single-mode control of a cantilever beam under principal parametric excitation, J. Sound Vib. 224(1), 33–47.
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.
Nayfeh  AH (1998), Reduced-order models of weakly nonlinear spatially continuous systems, Nonlinear Dyn. 16(2), 105–125.
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.
Chin  C-M and Nayfeh  AH (1997), Three-to-one internal resonances in hinged-clamped beams, Nonlinear Dyn. 12(2), 129–154.
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.
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.
Haxton  RS and Barr  ADS (1972), The autoparametric vibration absorber, ASME J. Eng. Ind. 94, 119–125.
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.
Hatwal  H, Mallik  AK, and Ghosh  A (1982), Nonlinear vibrations of a harmonically excited autoparametric system, J. Sound Vib. 81(2), 153–164.
Tondl A, Ruijgrok T, Verhulst F, and Nabergoj R (2000), Autoparametric Resonance in Mechanical Systems, Cambridge Univ Press, Cambridge, UK.
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.
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.
Lee  WK and Park  HD (1997), Chaotic dynamics of a harmonically excited spring-pendulum system with internal resonance, Nonlinear Dyn. 14(3), 211–229.
Balachandran  B and Khan  KA (1996), Spectral analysis of nonlinear interactions, Mech. Syst. Signal Process. 10(6), 711–727.
Kondou  T and Yagasaki  K (1995), Some recent topics on nonlinear vibration and chaos, Trans. Jpn. Soc. Mech. Eng., Ser. C 61, 746–751.
Iwan  WD and Stahl  KJ (1973), The response of an elastic disk with a moving mass system, ASME J. Appl. Mech. 40, 445–451.
Mote  CD (1970), Stability of circular plates subjected to moving loads, J. Franklin Inst. 290, 329–344.
Shen  IY (1993), Response of a stationary, damped, circular plate under a rotating slider bearing system, ASME J. Vibr. Acoust. 115(1), 65–69.
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.
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.
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.
Wiercigroch M and de Kraker B (eds), (2000), Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, World Scientific, 444.
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.
Heuer  R (1994), Large flexural vibrations of thermally stressed layered shallow shells, Nonlinear Dyn. 5(1), 25–38.
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.
Merchant  ME (1944), Basic mechanics in the metal cutting process, Trans. ASME 11, 168–175.
Grabec  I (1986), Chaos generated by the cutting process, Phys. Lett. A 117(8), 384–386.
Wiercigroch  M (1997), Chaotic vibration of a simple model of the machine tool cutting process system, ASME J. Vibr. Acoust. 119(3), 468–475.
Warmiński 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.
Warmiński  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.
Ganesan  R (1996), Dynamic response and stability of a rotor-support system with non-symmetric bearing clearances, Mech. Mach. Theory 31(6), 781–798.
Moon  J and Wickert  JA (1997), Nonlinear vibration of power transmission belts, J. Sound Vib. 200(4), 419–431.
Bogoliubov NN and Mitropolski YA (1961), Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York.
Zhang  L and Zu  JW (1998), Nonlinear vibrations of viscoelastic moving belts, Part 2: forced vibration analysis, J. Sound Vib. 216(1), 93–105.
Love AEH (1944), A Treatise on the Mathematical Theory of Elasticity, 4th Edition, 1st American Press, New York.
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.
Guckenheimer J and Holmes P (1983), Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York.
Kar  RC and Sujata  T (1990), Parametric instability of an elastically restrained cantilever beam, Comput. Struct. 34(3), 469–475.
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.
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.
Chin  C-M and Nayfeh  AH (1999), Three-to-one internal resonances in parametrically excited hinged-clamped beams, Nonlinear Dyn. 20(2), 131–158.
Cederbaum  G (1992), Analysis of parametrically excited laminated shells, Int. J. Mech. Sci. 34(3), 241–250.
Young  TH and Chen  FY (1993), Stability of fluttered panels subjected to in-plane harmonic forces, AIAA J. 31(9), 1667–1673.
Hansen  MH (2000), Effect of high frequency excitation on natural frequencies of spinning disks, J. Sound Vib. 234(4), 577–589.
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.
Meirovitch L (1970), Methods of Analytical Dynamics, McGraw-Hill, New York.
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.
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.
Asfar  KR and Masoud  KK (1994), Damping of parametrically excited single-degree-of-freedom systems, Int. J. Non-Linear Mech. 29(3), 421–428.
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.
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.
Tcherniak  D (1999), The influence of fast excitation on a continuous system, J. Sound Vib. 227(2), 343–360.
Fidlin  A (2000), On asymptotic properties of systems with strong and very strong high frequency excitation, J. Sound Vib. 235(2), 219–233.
Thomsen  JJ (1995), Chaotic dynamics of the partially follower-loaded elastic double pendulum, J. Sound Vib. 188(3), 385–405.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Ziegler  SW and Cartmell  MP (2001), Using motorized tethers for payload orbital transfer, J. Spacecr. Rockets 38(6), 904–913.
Ziegler SW (2003), Rigid body dynamics of tethers in space, PhD Thesis, Univ of Glasgow, Scotland, UK.
Chobotov VA (1996), Orbital mechanics, Orbital Mechanics, 2nd Edition, AIAA, Reston VA.  

Figures

Grahic Jump Location
Geometry of tethered dumbbell orbiting Earth
Grahic Jump Location
In-plane angular response between 4th and 5th orbit to initial displacement of ψ(0)=α(0)=0.1 rad (solid—numerical, dash—2nd order approx, chain dash—3rd order approx)
Grahic Jump Location
Out-of-plane angular response between 4th and 5th orbit to initial displacement of ψ(0)=α(0)=0.1 rad (solid—numerical, dash—2nd order approx, chain dash—3rd order approx)

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