메뉴 건너뛰기




Volumn 36, Issue 2, 2013, Pages 520-539

On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems

Author keywords

Invariant manifold; Nonlinear normal modes; Nonlinear vibrations; Numerical computation; Reduced order models

Indexed keywords

INVARIANT MANIFOLDS; NON-LINEAR VIBRATIONS; NONLINEAR NORMAL MODES; NUMERICAL COMPUTATIONS; REDUCED ORDER MODELS;

EID: 84875260309     PISSN: 08883270     EISSN: 10961216     Source Type: Journal    
DOI: 10.1016/j.ymssp.2012.10.016     Document Type: Article
Times cited : (30)

References (58)
  • 1
    • 85024529756 scopus 로고
    • The normal modes of nonlinear n-degree-of-freedom systems
    • R.M. Rosenberg The normal modes of nonlinear n-degree-of-freedom systems J. Appl. Mech. 29 1962 7 14
    • (1962) J. Appl. Mech. , vol.29 , pp. 7-14
    • Rosenberg, R.M.1
  • 2
    • 33645049419 scopus 로고
    • On non-linear vibrations of systems with many degrees of freedom
    • R.M. Rosenberg On non-linear vibrations of systems with many degrees of freedom Adv. Appl. Mech. 9 1966 155 242
    • (1966) Adv. Appl. Mech. , vol.9 , pp. 155-242
    • Rosenberg, R.M.1
  • 3
    • 0029388116 scopus 로고
    • Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems
    • N. Boivin, C. Pierre, and S. Shaw Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems Nonlinear Dyn. 8 1995 315 346
    • (1995) Nonlinear Dyn. , vol.8 , pp. 315-346
    • Boivin, N.1    Pierre, C.2    Shaw, S.3
  • 4
    • 0037262542 scopus 로고    scopus 로고
    • Fitting measured frequency response using non-linear normal modes
    • C. Gibert Fitting measured frequency response using non-linear normal modes Mech. Syst. Signal Process. 17 1 2003 211 218
    • (2003) Mech. Syst. Signal Process. , vol.17 , Issue.1 , pp. 211-218
    • Gibert, C.1
  • 6
    • 53849093038 scopus 로고    scopus 로고
    • Non-linear normal modes, part i a useful framework for the structural dynamicist
    • G. Kerschen, M. Peeters, J.C. Golinval, and A.F. Vakakis Non-linear normal modes, part I a useful framework for the structural dynamicist Mech. Syst. Signal Process. 23 1 2009 170 194
    • (2009) Mech. Syst. Signal Process. , vol.23 , Issue.1 , pp. 170-194
    • Kerschen, G.1    Peeters, M.2    Golinval, J.C.3    Vakakis, A.F.4
  • 7
    • 63549092696 scopus 로고    scopus 로고
    • Complex non-linear modal analysis for mechanical systems application to turbomachinery bladings with friction interface
    • D. Laxalde, and F. Thouverez Complex non-linear modal analysis for mechanical systems application to turbomachinery bladings with friction interface J. Sound Vib. 322 4-5 2009 1009 1025
    • (2009) J. Sound Vib. , vol.322 , Issue.45 , pp. 1009-1025
    • Laxalde, D.1    Thouverez, F.2
  • 8
    • 0027911991 scopus 로고
    • Normal modes for non-linear vibratory systems
    • S.W. Shaw, and C. Pierre Normal modes for non-linear vibratory systems J. Sound Vib. 164 1 1993 85 124
    • (1993) J. Sound Vib. , vol.164 , Issue.1 , pp. 85-124
    • Shaw, S.W.1    Pierre, C.2
  • 9
    • 0028197991 scopus 로고
    • On nonlinear normal modes of continuous systems
    • A.H. Nayfeh, and S.A. Nayfeh On nonlinear normal modes of continuous systems Trans. ASME/J. Vib. Acoust. 116 1994 129 136
    • (1994) Trans. ASME/J. Vib. Acoust. , vol.116 , pp. 129-136
    • Nayfeh, A.H.1    Nayfeh, S.A.2
  • 10
    • 0032097205 scopus 로고    scopus 로고
    • Reduced-order models of weakly non-linear spatially continuous systems
    • A.H. Nayfeh Reduced-order models of weakly non-linear spatially continuous systems Nonlinear Dyn. 16 1998 105 125
    • (1998) Nonlinear Dyn. , vol.16 , pp. 105-125
    • Nayfeh, A.H.1
  • 12
    • 0011646339 scopus 로고    scopus 로고
    • Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature
    • G. Rega, W. Lacarbonara, and A.H. Nayfeh Reduction methods for nonlinear vibrations of spatially continuous systems with initial curvature Solid Mech. Appl. 77 2000 235 246
    • (2000) Solid Mech. Appl. , vol.77 , pp. 235-246
    • Rega, G.1    Lacarbonara, W.2    Nayfeh, A.H.3
  • 13
    • 2342517458 scopus 로고    scopus 로고
    • Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes
    • C. Touzé, O. Thomas, and A. Chaigne Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes J. Sound Vib. 273 1-2 2004 77 101
    • (2004) J. Sound Vib. , vol.273 , Issue.12 , pp. 77-101
    • Touzé, C.1    Thomas, O.2    Chaigne, A.3
  • 14
    • 10044269430 scopus 로고    scopus 로고
    • Asymptotic non-linear normal modes for large amplitude vibrations of continuous structures
    • C. Touzé, O. Thomas, and A. Huberdeau Asymptotic non-linear normal modes for large amplitude vibrations of continuous structures Comput. Struct. 82 31-32 2004 2671 2682
    • (2004) Comput. Struct. , vol.82 , Issue.3132 , pp. 2671-2682
    • Touzé, C.1    Thomas, O.2    Huberdeau, A.3
  • 15
    • 18444364818 scopus 로고    scopus 로고
    • Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds
    • S.C. Sinha, S. Redkar, and E.A. Butcher Order reduction of nonlinear systems with time periodic coefficients using invariant manifolds J. Sound Vib. 284 3-5 2005 985 1002
    • (2005) J. Sound Vib. , vol.284 , Issue.35 , pp. 985-1002
    • Sinha, S.C.1    Redkar, S.2    Butcher, E.A.3
  • 16
    • 34547131037 scopus 로고    scopus 로고
    • Reduced-order models for non-linear vibrations of fluid-filled circular cylindrical shells comparison of pod and asymptotic non-linear normal modes methods
    • M. Amabili, and C. Touzé Reduced-order models for non-linear vibrations of fluid-filled circular cylindrical shells comparison of pod and asymptotic non-linear normal modes methods J. Fluids Struct. 23 6 2007 885 903
    • (2007) J. Fluids Struct. , vol.23 , Issue.6 , pp. 885-903
    • Amabili, M.1    Touzé, C.2
  • 17
    • 40349092905 scopus 로고    scopus 로고
    • Reduced-order models for large-amplitude vibrations of shells including in-plane inertia
    • C. Touzé, M. Amabili, and O. Thomas Reduced-order models for large-amplitude vibrations of shells including in-plane inertia Comput. Methods Appl. Mech. Eng. 197 21-24 2008 2030 2045
    • (2008) Comput. Methods Appl. Mech. Eng. , vol.197 , Issue.2124 , pp. 2030-2045
    • Touzé, C.1    Amabili, M.2    Thomas, O.3
  • 18
    • 67349212595 scopus 로고    scopus 로고
    • Nonlinear modes of clarinet-like musical instruments
    • D. Noreland, S. Bellizzi, C. Vergez, and R. Bouc Nonlinear modes of clarinet-like musical instruments J. Sound Vib. 324 3-5 2009 983 1002
    • (2009) J. Sound Vib. , vol.324 , Issue.35 , pp. 983-1002
    • Noreland, D.1    Bellizzi, S.2    Vergez, C.3    Bouc, R.4
  • 19
    • 0038005081 scopus 로고    scopus 로고
    • Dynamics of linear discrete systems connected to local, essentially non-linear attachments
    • A.F. Vakakis, L.I. Manevitch, O. Gendelman, and L. Bergman Dynamics of linear discrete systems connected to local, essentially non-linear attachments J. Sound Vib. 264 2003 559 577
    • (2003) J. Sound Vib. , vol.264 , pp. 559-577
    • Vakakis, A.F.1    Manevitch, L.I.2    Gendelman, O.3    Bergman, L.4
  • 20
    • 27544508422 scopus 로고    scopus 로고
    • Energy transfers in a system of two coupled oscillators with essential nonlinearity 1:1 resonance manifold and transient bridging orbits
    • G. Kerschen, A.F. Vakakis, Y.S. Lee, D.M. McFarland, J.J. Kowtko, and L.A. Bergman Energy transfers in a system of two coupled oscillators with essential nonlinearity 1:1 resonance manifold and transient bridging orbits Nonlinear Dyn. 42 2005 283 303
    • (2005) Nonlinear Dyn. , vol.42 , pp. 283-303
    • Kerschen, G.1    Vakakis, A.F.2    Lee, Y.S.3    McFarland, D.M.4    Kowtko, J.J.5    Bergman, L.A.6
  • 21
    • 18844398348 scopus 로고    scopus 로고
    • Energy pumping for a larger span of energy
    • E. Gourdon, and C.H. Lamarque Energy pumping for a larger span of energy J. Sound Vib. 285 2005 711 720
    • (2005) J. Sound Vib. , vol.285 , pp. 711-720
    • Gourdon, E.1    Lamarque, C.H.2
  • 25
    • 0015465649 scopus 로고
    • On periodic solutions close to rectilinear normal vibration modes
    • L.I. Manevitch, and Yu.V. Mikhlin On periodic solutions close to rectilinear normal vibration modes J. Appl. Math. Mech. (PMM) 36 6 1972 1051 1058
    • (1972) J. Appl. Math. Mech. (PMM) , vol.36 , Issue.6 , pp. 1051-1058
    • Manevitch, L.I.1    Mikhlin, Yu.V.2
  • 26
    • 0000809158 scopus 로고
    • A direct method for non-linear normal modes
    • R.H. Rand A direct method for non-linear normal modes Int. J. Non-linear Mech. 9 1974 363 368
    • (1974) Int. J. Non-linear Mech. , vol.9 , pp. 363-368
    • Rand, R.H.1
  • 27
    • 0028461550 scopus 로고
    • Energy-based formulation for computing nonlinear normal modes in undamped continuous systems
    • M.E. King, and A.F. Vakakis Energy-based formulation for computing nonlinear normal modes in undamped continuous systems J. Vib. Acoust. 116 1994 332 340
    • (1994) J. Vib. Acoust. , vol.116 , pp. 332-340
    • King, M.E.1    Vakakis, A.F.2
  • 28
    • 0030230830 scopus 로고    scopus 로고
    • Normal vibrations of a general class of conservative oscillators
    • Yu.V. Mikhlin Normal vibrations of a general class of conservative oscillators Non-linear Dyn. 11 1996 1 15
    • (1996) Non-linear Dyn. , vol.11 , pp. 1-15
    • Mikhlin, Yu.V.1
  • 29
    • 0026420003 scopus 로고
    • Non-linear normal modes and invariant manifolds
    • S.W. Shaw, and C. Pierre Non-linear normal modes and invariant manifolds J. Sound Vib. 150 1 1991 170 173
    • (1991) J. Sound Vib. , vol.150 , Issue.1 , pp. 170-173
    • Shaw, S.W.1    Pierre, C.2
  • 32
    • 0000077159 scopus 로고
    • Solutions d'un système d'équations différentielles dans le voisinage de valeurs singulières
    • H. Dulac Solutions d'un système d'équations différentielles dans le voisinage de valeurs singulières Bulletin de la Société Mathématique de France 40 1912 324 383
    • (1912) Bulletin de la Société Mathématique de France , vol.40 , pp. 324-383
    • Dulac, H.1
  • 33
    • 0026223029 scopus 로고
    • Analysis of non-linear dynamical systems by the normal form theory
    • L. Jézéquel, and C.H. Lamarque Analysis of non-linear dynamical systems by the normal form theory J. Sound Vib. 149 3 1991 429 459
    • (1991) J. Sound Vib. , vol.149 , Issue.3 , pp. 429-459
    • Jézéquel, L.1    Lamarque, C.H.2
  • 34
    • 33748795371 scopus 로고    scopus 로고
    • Non-linear normal modes for damped geometrically non-linear systems application to reduced-order modeling of harmonically forced structures
    • C. Touzé, and M. Amabili Non-linear normal modes for damped geometrically non-linear systems application to reduced-order modeling of harmonically forced structures J. Sound Vib. 298 4-5 2006 958 981
    • (2006) J. Sound Vib. , vol.298 , Issue.45 , pp. 958-981
    • Touzé, C.1    Amabili, M.2
  • 35
    • 0030780655 scopus 로고    scopus 로고
    • Non-linear normal modes (nnms) and their application in vibration theory an overview
    • A.F. Vakakis Non-linear normal modes (nnms) and their application in vibration theory an overview Mech. Syst. Signal Process. 11 1 1997 3 22
    • (1997) Mech. Syst. Signal Process. , vol.11 , Issue.1 , pp. 3-22
    • Vakakis, A.F.1
  • 36
    • 14744299273 scopus 로고    scopus 로고
    • Geodesics, nonlinear normal modes of conservative vibratory systems and decomposition method
    • X. Zhang Geodesics, nonlinear normal modes of conservative vibratory systems and decomposition method J. Sound Vib. 282 2005 971 989
    • (2005) J. Sound Vib. , vol.282 , pp. 971-989
    • Zhang, X.1
  • 37
    • 0022191004 scopus 로고
    • The calculation of strongly nonlinear systems close to vibration-impact systems
    • V.N. Pilipchuk The calculation of strongly nonlinear systems close to vibration-impact systems Prikl. Mat. Mech. 49 5 1985 572 578
    • (1985) Prikl. Mat. Mech. , vol.49 , Issue.5 , pp. 572-578
    • Pilipchuk, V.N.1
  • 38
    • 0002900162 scopus 로고
    • A transformation for vibrating systems based on a non-smooth periodic pair of functions
    • (in Russian)
    • V.N. Pilipchuk A transformation for vibrating systems based on a non-smooth periodic pair of functions Dokl. An. Ukr. SSR 4 1988 37 40 (in Russian)
    • (1988) Dokl. An. Ukr. SSR , vol.4 , pp. 37-40
    • Pilipchuk, V.N.1
  • 39
    • 0006174897 scopus 로고    scopus 로고
    • Study of a class of subharmonic motions using a non-smooth temporal transformation (nstt)
    • V.N. Pilipchuk, A.F. Vakakis, and M.A.F. Azeez Study of a class of subharmonic motions using a non-smooth temporal transformation (nstt) Physica D 100 1997 145 164
    • (1997) Physica D , vol.100 , pp. 145-164
    • Pilipchuk, V.N.1    Vakakis, A.F.2    Azeez, M.A.F.3
  • 40
    • 18844436998 scopus 로고    scopus 로고
    • Complicated dynamics of a linear oscillator with a light, essentially non-linear attachment
    • Y.S. Lee, G. Kerschen, A.F. Vakakis, P. Panagopoulos, L. Bergman, and D.M. McFarland Complicated dynamics of a linear oscillator with a light, essentially non-linear attachment Physica D 204 2005 41 69
    • (2005) Physica D , vol.204 , pp. 41-69
    • Lee, Y.S.1    Kerschen, G.2    Vakakis, A.F.3    Panagopoulos, P.4    Bergman, L.5    McFarland, D.M.6
  • 41
    • 0030151935 scopus 로고    scopus 로고
    • A numerical method for determining nonlinear normal modes
    • J.C. Slater A numerical method for determining nonlinear normal modes Nonlinear Dyn. 10 1 1996 19 30
    • (1996) Nonlinear Dyn. , vol.10 , Issue.1 , pp. 19-30
    • Slater, J.C.1
  • 43
    • 33746882845 scopus 로고    scopus 로고
    • Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes
    • R. Arquier, S. Bellizzi, R. Bouc, and B. Cochelin Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes Comput. Struct. 84 24-25 2006 1565 1576
    • (2006) Comput. Struct. , vol.84 , Issue.2425 , pp. 1565-1576
    • Arquier, R.1    Bellizzi, S.2    Bouc, R.3    Cochelin, B.4
  • 44
    • 53849083425 scopus 로고    scopus 로고
    • Non-linear normal modes, part II toward a practical computation using numerical continuation techniques
    • M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, and J.C. Golinval Non-linear normal modes, part II toward a practical computation using numerical continuation techniques Mech. Syst. Signal Process. 23 1 2009 195 216
    • (2009) Mech. Syst. Signal Process. , vol.23 , Issue.1 , pp. 195-216
    • Peeters, M.1    Viguié, R.2    Sérandour, G.3    Kerschen, G.4    Golinval, J.C.5
  • 46
    • 67349260182 scopus 로고    scopus 로고
    • A high-order purely frequency-based harmonic balance formulation for continuation of periodic solutions
    • B. Cochelin, and C. Vergez A high-order purely frequency-based harmonic balance formulation for continuation of periodic solutions J. Sound Vib. 324 1-2 2009 243 262
    • (2009) J. Sound Vib. , vol.324 , Issue.12 , pp. 243-262
    • Cochelin, B.1    Vergez, C.2
  • 48
    • 8744230577 scopus 로고    scopus 로고
    • Matcont a MATLAB package for numerical bifurcation analysis of ODEs
    • A. Dhooge, W. Govaerts, and Y.A. Kuznetsov Matcont a MATLAB package for numerical bifurcation analysis of ODEs ACM Trans. Math. Software 29 2 2003 141 164 Available at: 〈 http://sourceforge.net/projects/matcont/ 〉
    • (2003) ACM Trans. Math. Software , vol.29 , Issue.2 , pp. 141-164
    • Dhooge, A.1    Govaerts, W.2    Kuznetsov, Y.A.3
  • 49
    • 0028203123 scopus 로고
    • Normal modes of vibration for non-linear continuous systems
    • S.W. Shaw, and C. Pierre Normal modes of vibration for non-linear continuous systems J. Sound Vib. 169 3 1994 85 124
    • (1994) J. Sound Vib. , vol.169 , Issue.3 , pp. 85-124
    • Shaw, S.W.1    Pierre, C.2
  • 50
    • 0037474503 scopus 로고    scopus 로고
    • A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds
    • E. Pesheck, C. Pierre, and S. Shaw A new Galerkin-based approach for accurate non-linear normal modes through invariant manifolds J. Sound Vib. 249 5 2002 971 993
    • (2002) J. Sound Vib. , vol.249 , Issue.5 , pp. 971-993
    • Pesheck, E.1    Pierre, C.2    Shaw, S.3
  • 51
    • 2142657195 scopus 로고    scopus 로고
    • Large-amplitude non-linear normal modes of piecewise linear systems
    • D. Jiang, C. Pierre, and S. Shaw Large-amplitude non-linear normal modes of piecewise linear systems J. Sound Vib. 272 3-5 2004 869 891
    • (2004) J. Sound Vib. , vol.272 , Issue.35 , pp. 869-891
    • Jiang, D.1    Pierre, C.2    Shaw, S.3
  • 52
    • 26444518712 scopus 로고    scopus 로고
    • Nonlinear normal modes for vibratory systems under harmonic excitation
    • D. Jiang, C. Pierre, and S. Shaw Nonlinear normal modes for vibratory systems under harmonic excitation J. Sound Vib. 288 4-5 2005 791 812
    • (2005) J. Sound Vib. , vol.288 , Issue.45 , pp. 791-812
    • Jiang, D.1    Pierre, C.2    Shaw, S.3
  • 53
    • 13544274198 scopus 로고    scopus 로고
    • The construction of non-linear normal modes for systems with internal resonance
    • D. Jiang, C. Pierre, and S. Shaw The construction of non-linear normal modes for systems with internal resonance Int. J. Non-linear Mech. 40 5 2005 729 746
    • (2005) Int. J. Non-linear Mech. , vol.40 , Issue.5 , pp. 729-746
    • Jiang, D.1    Pierre, C.2    Shaw, S.3
  • 54
    • 84875249536 scopus 로고    scopus 로고
    • Computation of nonlinear normal modes of large structures: Application to a full-scale aircraft
    • Roma
    • M. Peeters, L. Renson, G. Kerschen, J.-C. Golinval, C. Stéphan, P. Lubrina, Computation of nonlinear normal modes of large structures: application to a full-scale aircraft, in: Proceedings of the Seventh Nonlinear Dynamics Conference, ENOC-2011, Roma, 2011.
    • (2011) Proceedings of the Seventh Nonlinear Dynamics Conference, ENOC-2011
    • Peeters, M.1
  • 57
    • 84871017828 scopus 로고    scopus 로고
    • An upper bound for validity limits of asymptotic analytical approaches based on normal form theory, Non-linear Dynamics
    • C.-H. Lamarque, C. Touzé, O. Thomas, An upper bound for validity limits of asymptotic analytical approaches based on normal form theory, Non-linear Dynamics, Non-linear Dyn. 70 (3) (2012) 1931-1949.
    • (2012) Non-linear Dyn. , vol.70 , Issue.3 , pp. 1931-1949
    • Lamarque, H.C.1
  • 58
    • 67649202516 scopus 로고    scopus 로고
    • Modal analysis of a nonlinear periodic structure with cyclic symmetry
    • F. Georgiades, M. Peeters, G. Kerschen, J.C. Golinval, and M. Ruzzene Modal analysis of a nonlinear periodic structure with cyclic symmetry AIAA J. 47 2009 1014 1025
    • (2009) AIAA J. , vol.47 , pp. 1014-1025
    • Georgiades, F.1    Peeters, M.2    Kerschen, G.3    Golinval, J.C.4    Ruzzene, M.5


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.