Skip to main content
SpringerLink
Log in

Two Formulations of Nonlinear Normal Vibration Modes and Their Applications

  • Conference paper
  • First Online:
IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design

Part of the book series: IUTAM Bookseries (closed) ((IUTAMBOOK,volume 32))

  • 1363 Accesses

Abstract

Two formulations of nonlinear normal vibration modes (NNMs) in conservative and near conservative systems are considered. Construction of the NNMs and their applications in some mechanical problems are presented. Namely, the nonlinear vibro-absorption problem, the cylindrical shell nonlinear dynamics and the vehicle suspension nonlinear dynamics are analyzed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Avramov, K.V., Mikhlin, Y.V.: Snap- through truss as a vibration absorber. J. Vib. Control. 10, 291–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Avramov, K.V., Mikhlin, Y.V.: Snap-through truss as an absorber of forced oscillations. J. Sound Vib. 29, 705–722 (2006)

    Article  Google Scholar 

  • Avramov, K.V., Mikhlin, Y.V., Kurilov, E.: Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells. Nonlinear Dyn. 47, 331–352 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Bolotin, V.V.: Non-Conservative Problems of the Theory of Elastic Stability. Macmillan, New York (1963)

    Google Scholar 

  • Budiansky, B., Hutchinson, J.W.: Dynamic buckling of imperfect-sensitive structures. Appl. Mech., pp. 636–657. Springer, Berlin (1966)

    Google Scholar 

  • Fung, Y.C., Sechler, E.E. (eds.): Thin- Shell Structures, Theory, Experiment and Design. Prentice-Hall, Englewood Cliff (1974)

    Google Scholar 

  • Kauderer, H.: Nichtlineare Mechanik. Springer, Berlin (1958)

    Book  MATH  Google Scholar 

  • King, M.E., Vakakis, A.F.: An energy-based formulation for computing nonlinear normal modes in undamped continuous systems. J. Vib. Acoust. 116(3), 332–340 (1993)

    Article  Google Scholar 

  • Kinney, W., Rosenberg, R.: On the steady state vibrations of nonlinear systems with many degrees of freedom. J. Appl. Mech. 33, 406–412 (1966)

    Article  MATH  Google Scholar 

  • Kubenko, V.D., Koval’shuk, P.S., Krasnopolskaya, T.S.: Nonlinear Interaction of Modes of Bending Shell Oscillations. Naukova Dumka, Kiev (1984) (in Russian)

    Google Scholar 

  • Manevich, L.I., Mikhlin, Y.V.: Periodic solutions close to rectilinear normal vibration modes. Prikl. Mat. i Mekhanica (PMM USSR) 36, 1051–1058 (1972)

    MathSciNet  Google Scholar 

  • Manevich, L., Mikhlin, Y., Pilipchuk, V.: The Method of Normal Oscillation for Essentially Nonlinear Systems. Nauka, Moscow (1989) (in Russian)

    Google Scholar 

  • Mikhlin, Y.: Resonance modes of near-conservative nonlinear systems. Prikl. Mat. I Mekhanica (PMM USSR) 38, 425–429 (1974)

    MATH  Google Scholar 

  • Mikhlin, Y.: Matching of local expansions in the theory of nonlinear vibrations. J. Sound Vib. 182, 577–588 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  • Mikhlin, Y.: Normal vibrations of a general class of conservative oscillators. Nonlinear Dyn. 11, 1–16 (1996)

    Article  MathSciNet  Google Scholar 

  • Mikhlin, Y., Morgunov, B.: Normal vibrations in near-conservative self-exited and viscoelastic nonlinear systems. Nonlinear Dyn. 25, 33–48 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  • Mikhlin, Y.V., Reshetnikova, S.N.: Dynamical interaction of an elastic system and essentially nonlinear absorber. J. Sound Vib. 283, 91–120 (2005)

    Article  MATH  Google Scholar 

  • Mikhlin, Yu.V., Avramov, K.V.: Nonlinear normal modes of vibrating mechanical systems. Review of theoretical developments. Appl. Mech. Rev. (2011, in press)

    Google Scholar 

  • Nayfeh, A., Nayfeh, S.: On nonlinear modes of continuous systems. J. Vib. Acoust. 116, 129–136 (1994)

    Article  Google Scholar 

  • Pilipchuk, V.N.: Impact modes in discrete vibrating systems with rigid barriers. Int. J. Non-Linear Mech. 36, 999–1012 (2001)

    Article  MATH  Google Scholar 

  • Pilipchuk, V.N.: Nonlinear Dynamics. Between Linear and Impact Limits. Springer, Berlin (2010)

    MATH  Google Scholar 

  • Rauscher, M.: Steady oscillations of system with nonlinear and unsymmetrical elasticity. J. Appl. Mech. 5, A-169 (1938)

    Google Scholar 

  • Rosenberg, R.M.: The normal modes of nonlinear n-degrees-of-freedom systems. J. Appl. Mech. 30, 7–14 (1962)

    Article  Google Scholar 

  • Rosenberg, R.M.: Nonlinear vibrations of systems with many degrees of freedom. In: Advances of Applied Mechanics 9, pp. 156–243. Academic Press, New York (1966)

    Google Scholar 

  • Shaw, S., Pierre, C.: Nonlinear normal modes and invariant manifolds. J. Sound Vib. 150, 170–173 (1991)

    Article  MathSciNet  Google Scholar 

  • Shaw, S., Pierre, C.: Normal modes for nonlinear vibratory systems. J. Sound Vib. 164, 85–124 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Shaw, S., Pierre, C.: Normal modes of vibration for non-linear continuous systems. J. Sound Vib. 169(3), 319–347 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  • Vakakis, A.F., Cetinkaya, C.: Mode localization in a class of multi-degree-of-freedom systems with cyclic symmetry. SIAM J. Appl. Math. 53, 265–282 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Vakakis, A.F., Rand, R.: Normal modes and global dynamics of a two-degree-of freedom non-linear system. I. Low energies. II. High energies. Int. J. Non-Linear Mech. 27, 861–888 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Vakakis, A., Manevitch, L., Mikhlin, Y., Pilipchuk, V., Zevin, A.: Normal Modes and Localization in Nonlinear Systems. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  • Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems. Springer, Berlin (2008)

    Google Scholar 

Download references

Acknowledgments

This work was supported in part by a grant M9403 ДР 0109U002426 from the Ministry of Science and Education of Ukraine and by a grant Ф28/257 from the National Academy of Science of Ukraine and Russian Foundation for Basic Research.

Author information

Authors and Affiliations

  1. National Technical University “Kharkov Polytechnic Institute”, 21 Frunze str, Kharkov, 61002, Ukraine

    Yuri V. Mikhlin

Authors
  1. Yuri V. Mikhlin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yuri V. Mikhlin .

Editor information

Editors and Affiliations

  1. College of Physical Sciences, Department of Engineering, Centre for, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

    Marian Wiercigroch

  2. Structural and Geotechnical Engineering, Sapienza University of Rome, Via A. Gramsci 53, Roma, 00197, Italy

    Giuseppe Rega

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media Dordrecht

About this paper

Cite this paper

Mikhlin, Y.V. (2013). Two Formulations of Nonlinear Normal Vibration Modes and Their Applications. In: Wiercigroch, M., Rega, G. (eds) IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design. IUTAM Bookseries (closed), vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5742-4_3

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-5742-4_3

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-5741-7

  • Online ISBN: 978-94-007-5742-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation