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Nonlinear Normal Modes and Reduced Order Models

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Lectures on Nonlinear Dynamics

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

Nonlinear normal modes (NNMs) have been used in the last decades as an additional tool for the analysis of multi degree of freedom (mdof) dynamical systems and for the derivation of reduced order models. Examples in literature show that only a few NNMs are usually required to model, in many problems, their forced dynamics. In this chapter a brief literature review with the seminal contributions in this field and recent contributions are presented. Then the definitions of NNMs based on the works by Rosenberg and Shaw and Pierre are given. According to the latter definition, a nonlinear vibration mode is a two-dimensional invariant manifold in the phase space of the system. This is approximated by asymptotic expansions in terms of a pair of master coordinates or through a Galerkin expansion. This methodology and additional numerical tools are presented here together with didactical examples that illustrate some unique characteristics of NNMs as compared to linear normal modes. Also Poincaré sections are used for the identification of NNMs and their stability and continuation methods are used to obtain frequency-energy plots and frequency-amplitude relations. The concept of multi-mode is briefly presented.

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References

  1. Apiwattanalunggarn, P.: Model reduction of nonlinear structural systems using nonlinear normal modes and component mode synthesis. Ph.D. thesis, Univesity of Michigan (2003)

    Google Scholar 

  2. Apiwattanalunggarn, P., Shaw, S.W., Pierre, C., Jiang, D.: Finite-element-based nonlinear modal reduction of a rotating beam with large-amplitude motion. Journal of Vibration and Control 9(3-4), 235–263 (2003). https://doi.org/10.1177/107754603030751

    Article  MathSciNet  MATH  Google Scholar 

  3. Avramov, K.V., Mikhlin, Y.V.: Review of applications of nonlinear normal modes for vibrating mechanical systems. Applied Mechanics Reviews 65(2) (2013). https://doi.org/10.1115/1.4023533

  4. Blanc, F., Touzé, C., Mercier, J.F., Ege, K., Bonnet-Ben Dhia, A.S.: On the numerical computation of nonlinear normal modes for reduced-order modelling of conservative vibratory systems. Mechanical Systems and Signal Processing 36(2), 520–539 (2013). https://doi.org/10.1016/j.ymssp.2012.10.016. URL https://hal.archives-ouvertes.fr/hal-00772317

  5. Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. In: AIAA Dynamics Specialists Conference, Hilton Head, SC, (1994)

    Google Scholar 

  6. Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear normal modes, invariance, and modal dynamics approximations of nonlinear systems. Nonlinear dynamics 8(3), 315–346 (1995)

    Article  MathSciNet  Google Scholar 

  7. Gavassoni, E.: Application of nonlinear vibration modes to conceptual models of offshore structures. Ph.D. thesis, PUC-Rio (2012)

    Google Scholar 

  8. Gavassoni, E., Gonçalves, P.B., Roehl, D.M.: Nonlinear vibration modes of an offshore articulated tower. Ocean Engineering 109, 226–242 (2015). https://doi.org/10.1016/j.oceaneng.2015.08.028

    Article  Google Scholar 

  9. Haller, G., Ponsioen, S.: Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction. Nonlinear Dynamics 86(3), 1493–1534 (2016). https://doi.org/10.1007/s11071-016-2974-z

    Article  MathSciNet  MATH  Google Scholar 

  10. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer New York (2013)

    Google Scholar 

  11. Jezequel, L., Lamarque, C.: Analysis of non-linear dynamical systems by the normal form theory. Journal of Sound and Vibration 149(3), 429–459 (1991). https://doi.org/10.1016/0022-460x(91)90446-q

    Article  Google Scholar 

  12. Jiang, D.: Nonlinear modal analysis based on invariant manifolds – application to rotating blade systems. Ph.D. thesis, Univesity of Michigan (2004)

    Google Scholar 

  13. Kerschen, G., Peeters, M., Golinval, J., Vakakis, A.: Nonlinear normal modes, part i: A useful framework for the structural dynamicist. Mechanical Systems and Signal Processing 23(1), 170–194 (2009). https://doi.org/10.1016/j.ymssp.2008.04.002

    Article  Google Scholar 

  14. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.: Nonlinear normal modes, part i: An attempt to demystify them. In: In: 26th International Modal Analysis Conference, Orlando (2008)

    Google Scholar 

  15. Kuether, R.J., Allen, M.S.: A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models. Mechanical Systems and Signal Processing 46(1), 1–15 (2014). https://doi.org/10.1016/j.ymssp.2013.12.010

    Article  Google Scholar 

  16. Lacarbonara, W., Rega, G., Nayfeh, A.: Resonant non-linear normal modes. part i: Analytical treatment for structural one-dimensional systems. International Journal of Nonlinear Mechanics 38(6), 851–872 (2003). https://doi.org/10.1016/s0020-7462(02)00033-1

  17. Manevich, L., Mikhlin, I.: On periodic solutions close to rectilinear normal vibration modes. Journal of Applied Mathematics and Mechanics 36(6), 988–994 (1972). https://doi.org/10.1016/0021-8928(72)90032-9

    Article  MathSciNet  MATH  Google Scholar 

  18. Mazzilli, C.E.N., Neto, O.B.: Evaluation of nonlinear normal modes for finite-element models. Computers & Structures 80(11), 957–965 (2002)

    Article  Google Scholar 

  19. Meirovitch, L.: Elements of vibration analysis. McGraw-Hill Science (1975)

    Google Scholar 

  20. Meyrand, L., Sarrouy, E., Cochelin, B., Ricciardi, G.: Nonlinear normal mode continuation through a proper generalized decomposition approach with modal enrichment. Journal of Sound and Vibration 443, 444–459 (2019). https://doi.org/10.1016/j.jsv.2018.11.030

    Article  Google Scholar 

  21. Mikhlin, Y.: Nonlinear normal vibration modes and their applications. In: In Proceedings of the 9th Brazilian conference on dynamics Control and their Applications, pp. 151–171 (2010)

    Google Scholar 

  22. Month, L.A., Rand, R.H.: An application of the Poincaré map to the stability of nonlinear normal modes. Journal of Applied Mechanics 47(3), 645–651 (1980). https://doi.org/10.1115/1.3153747

    Article  MathSciNet  MATH  Google Scholar 

  23. Nayfeh, A., Balachandran, B.: Applied nonlinear dynamics. Wiley, New York (1995)

    Book  MATH  Google Scholar 

  24. Nayfeh, A.H.: Nonlinear interactions: analytical, computational, and experimental methods. Wiley, New York (2000)

    MATH  Google Scholar 

  25. Nayfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. Applied Mechanics Reviews 42(11S), S175–S201 (1989). https://doi.org/10.1115/1.3152389

    Article  MathSciNet  MATH  Google Scholar 

  26. Nayfeh, A.H., Chin, C., Nayfeh, S.A.: On nonlinear normal modes of systems with internal resonance. Journal of Vibration and Acoustics 118(3), 340–345 (1996). https://doi.org/10.1115/1.2888188

    Article  Google Scholar 

  27. Nayfeh, A.H., Nayfeh, S.A.: On nonlinear modes of continuous systems. Journal of Vibration and Acoustics 116(1), 129–136 (1994). https://doi.org/10.1115/1.2930388

    Article  Google Scholar 

  28. Orlando, D.: Nonlinear dynamics, instability and control of structural systems with modal interaction. Ph.D. thesis, Pontifícia Universidade Católica do Rio de Janeiro (2010)

    Google Scholar 

  29. Orlando, D., Gonçalves, P.B., Rega, G., Lenci, S.: Influence of symmetries and imperfections on the non-linear vibration modes of archetypal structural systems. International Journal of Non-Linear Mechanics 49, 175–195 (2013). https://doi.org/10.1016/j.ijnonlinmec.2012.10.004

    Article  Google Scholar 

  30. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, part II: Toward a practical computation using numerical continuation techniques. Mechanical Systems and Signal Processing 23(1), 195–216 (2009). https://doi.org/10.1016/j.ymssp.2008.04.003

    Article  Google Scholar 

  31. Pesheck, E., Boivin, N., Pierre, C., Shaw, S.W.: Nonlinear modal analysis of structural systems using multi-mode invariant manifolds. Nonlinear Dynamics 25(1/3), 183–205 (2001). https://doi.org/10.1023/a:1012910918498

    Article  MathSciNet  MATH  Google Scholar 

  32. Pesheck, E., Pierre, C., Shaw, S.W.: A new Galerkin-based approach for accurate nonlinear normal modes through invariant manifolds. Journal of Sound and Vibration 249(5), 971–993 (2002). https://doi.org/10.1006/jsvi.2001.3914

    Article  MathSciNet  MATH  Google Scholar 

  33. Rand, R.H.: Nonlinear normal modes in two-degree-of-freedom systems. Journal of Applied Mechanics 38(2), 561–561 (1971). https://doi.org/10.1115/1.3408826

    Article  Google Scholar 

  34. Renson, L., Deliége, G., Kerschen, G.: An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems. Meccanica 49(8), 1901–1916 (2014). https://doi.org/10.1007/s11012-014-9875-3

    Article  MathSciNet  MATH  Google Scholar 

  35. Rosenberg, R.: On nonlinear vibrations of systems with many degrees of freedom. In: Advances in Applied Mechanics, pp. 155–242. Elsevier (1966). https://doi.org/10.1016/s0065-2156(08)70008-5

  36. Rosenberg, R.M.: Normal modes of nonlinear dual-mode systems. Journal of Applied Mechanics 27, 263–268 (1960)

    Article  MATH  Google Scholar 

  37. Shaw, S., Pierre, C.: Non-linear normal modes and invariant manifolds. Journal of Sound and Vibration 150(1), 170–173 (1991). https://doi.org/10.1016/0022-460x(91)90412-d

    Article  MathSciNet  Google Scholar 

  38. Shaw, S., Pierre, C.: Normal modes for non-linear vibratory systems. Journal of Sound and Vibration 164(1), 85–124 (1993). https://doi.org/10.1006/jsvi.1993.1198

    Article  MathSciNet  MATH  Google Scholar 

  39. Shaw, S., Pierre, C.: Normal modes of vibration for non-linear continuous systems. Journal of Sound and Vibration 169(3), 319–347 (1994). https://doi.org/10.1006/jsvi.1994.1021

    Article  MathSciNet  MATH  Google Scholar 

  40. Shaw, S.W., Pierre, C., Pesheck, E.: Modal analysis-based reduced-order models for nonlinear structures : An invariant manifold approach. The Shock and Vibration Digest 31, 3–16 (1999)

    Article  Google Scholar 

  41. Soares, M., Mazzilli, C.: Nonlinear normal modes of planar frames discretised by the finite element method. Computers & Structures 77(5), 485–493 (2000). https://doi.org/10.1016/s0045-7949(99)00233-3

    Article  Google Scholar 

  42. Thompson, J.M.T.: Elastic instability phenomena. John Wiley & Sons, Chichester New York (1984)

    MATH  Google Scholar 

  43. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures. Journal of Sound and Vibration 298(4-5), 958–981 (2006). https://doi.org/10.1016/j.jsv.2006.06.032

    Article  Google Scholar 

  44. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. Journal of Sound and Vibration 273(1-2), 77–101 (2004). https://doi.org/10.1016/j.jsv.2003.04.005

    Article  Google Scholar 

  45. Touzé, C., Vizzaccaro, A., Thomas, O.: Model order reduction methods for geometrically nonlinear structures: A review of nonlinear techniques. Nonlinear Dynamics 105(2), 1141–1190 (2021). https://doi.org/10.1007/s11071-021-06693-9

    Article  Google Scholar 

  46. Vakakis, A.: Nonlinear normal modes and their applications in vibration theory: An overview. Mechanical Systems and Signal Processing 11(1), 3–22 (1997). https://doi.org/10.1006/mssp.1996.9999

    Article  MathSciNet  Google Scholar 

  47. Vakakis, A.F.: Analysis and identification of linear and nonlinear normal modes in vibrating systems. Ph.D. thesis, California Institute of Technology (1991). https://doi.org/10.7907/60CA-1Q64

  48. Vakakis, A.F., Manevitch, L.I., Mikhlin, Y.V., Pilipchuk, V.N., Zevin, A.A.: Normal modes and localization in nonlinear systems. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  49. Wiggins, S.: Introduction to applied nonlinear dynamical systems and chaos. Springer, New York (2003)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Pontifical Catholic University of Rio de Janeiro, PUCRio, Rio de Janeiro, Brazil

    Paulo Batista Gonçalves

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  1. Paulo Batista Gonçalves
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Correspondence to Paulo Batista Gonçalves .

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Editors and Affiliations

  1. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    José Roberto Castilho Piqueira

  2. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Carlos Eduardo Nigro Mazzilli

  3. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Celso Pupo Pesce

  4. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Guilherme Rosa Franzini

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Gonçalves, P.B. (2024). Nonlinear Normal Modes and Reduced Order Models. In: Castilho Piqueira, J.R., Nigro Mazzilli, C.E., Pesce, C.P., Franzini, G.R. (eds) Lectures on Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-45101-0_4

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  • DOI: https://doi.org/10.1007/978-3-031-45101-0_4

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  • Online ISBN: 978-3-031-45101-0

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