Skip to main content
SpringerLink
Log in

Linear and nonlinear modal analysis of the axially moving continua based on the invariant manifold method

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

By studying the transverse dynamics of the axially moving beam, the linear and nonlinear complex modes of gyroscopic continua are investigated based on the invariant manifold method. The nonlinear partial differential equations are truncated into a set of ordinary differential equations by the Galerkin method. The invariant manifold method is then used to obtain the linear and nonlinear complex modes. The gyroscopic effect due to the axially moving speed is discussed by comparing the proportions of the trial functions in the complex mode during the modal motions. The ‘traveling wave’ phenomena are found for such a gyroscopic continuum. The power of the invariant manifold method in the application to gyroscopic systems is demonstrated and discussed by taking the axially moving continuum as an example.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Wickert, J.A.: Transient vibration of gyroscopic systems with unsteady superposed motion. J. Sound Vib. 195, 797–807 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Mote Jr., C.D.: A study of band saw vibrations. J. Frankl. Inst. 279, 430–444 (1965)

    Article  Google Scholar 

  3. Tabarrok, B., Leech, C.M., Kim, Y.I.: Dynamics of an axially moving beam. J. Frankl. I I(297), 201–220 (1974)

    Article  MATH  Google Scholar 

  4. Ulsoy, A.G., Mote, C.D., Szymani, R.: Principal developments in band saw vibration and stability research. Holz Roh Werkst. 36, 273–280 (1978)

    Article  Google Scholar 

  5. Marynowski, K., Kapitaniak, T.: Dynamics of axially moving continua. Int. J. Mech. Sci. 81, 26–41 (2014)

    Article  Google Scholar 

  6. Banerjee, J.R., Kennedy, D.: Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects. J. Sound Vib. 333, 7299–7312 (2014)

    Article  Google Scholar 

  7. Banerjee, J.R., Papkov, S.O., Liu, X., Kennedy, D.: Dynamic stiffness matrix of a rectangular plate for the general case. J. Sound Vib. 342, 177–199 (2015)

    Article  Google Scholar 

  8. Rao, J.S.: History of Rotating Machinery Dynamics. Springer, Netherlands (2011)

    Book  Google Scholar 

  9. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1. Elsevier Academic Press, London (1998)

    Google Scholar 

  10. Ibrahim, R.A.: Overview of mechanics of pipes conveying fluids-part I: fundamental studies. J. Press. Vessel Technol. 132, 034001 (2010)

    Article  Google Scholar 

  11. Yu, D.L., Paidoussis, M.P., Shen, H.J., Wang, L.: Dynamic stability of periodic pipes conveying fluid. J. Appl. Mech.-T ASME 81, 011008 (2014)

  12. Meirovitch, L.: A new method of solution of the eigenvalue problem for gyroscopic systems. AIAA J. 12, 1337–1342 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meirovitch, L.: A modal analysis for the response of linear gyroscopic systems. ASME J. Appl. Mech. 42, 446–450 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hughes, P.C., Deleuterio, G.M.T.: Modal parameter analysis of gyroelastic continua. J. Appl. Mech.-T ASME 53, 918–924 (1986)

    Article  MATH  Google Scholar 

  15. Perkins, N.C.: Linear dynamics of a translating string on an elastic-foundation. J. Vib. Acoust. 112, 2–7 (1990)

    Article  Google Scholar 

  16. Wickert, J.A., Mote, C.D.: Classical vibration analysis of axially moving continua. J. Appl. Mech.-T ASME 57, 738–744 (1990)

    Article  MATH  Google Scholar 

  17. Oz, H.R., Pakdemirli, M., Boyaci, H.: Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int. J. Non-Linear Mech. 36, 107–115 (2001)

    Article  MATH  Google Scholar 

  18. Chen, L.Q., Yang, X.D.: Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models. Int. J. Solids Struct. 42, 37–50 (2005)

    Article  MATH  Google Scholar 

  19. Mockensturm, E.M., Guo, J.: Nonlinear vibration of parametrically excited, viscoelastic, axially moving strings. J. Appl. Mech. 72, 374 (2005)

    Article  MATH  Google Scholar 

  20. Chen, S.H., Huang, J.L., Sze, K.Y.: Multidimensional Lindstedt–Poincaré method for nonlinear vibration of axially moving beams. J. Sound Vib. 306, 1–11 (2007)

    Article  Google Scholar 

  21. Ponomareva, S.V., Horssen, W.T.: On transversal vibrations of an axially moving string with a time-varying velocity. Nonlinear Dyn. 50, 315–323 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, L.-Q., Zu, J.W.: Solvability condition in multi-scale analysis of gyroscopic continua. J. Sound Vib. 309, 338–342 (2008)

    Article  Google Scholar 

  23. Pakdemirli, M., Öz, H.R.: Infinite mode analysis and truncation to resonant modes of axially accelerated beam vibrations. J. Sound Vib. 311, 1052–1074 (2008)

    Article  Google Scholar 

  24. Malookani, R.A., van Horssen, W.T.: On resonances and the applicability of Galerkin’s truncation method for an axially moving string with time-varying velocity. J. Sound Vib. 344, 1–17 (2015)

    Article  Google Scholar 

  25. Thomsen, J.J., Dahl, J.: Analytical predictions for vibration phase shifts along fluid-conveying pipes due to Coriolis forces and imperfections. J. Sound Vib. 329, 3065–3081 (2010)

    Article  Google Scholar 

  26. Čepon, G., Boltežar, M.: Computing the dynamic response of an axially moving continuum. J. Sound Vib. 300, 316–329 (2007)

    Article  Google Scholar 

  27. Ding, H., Chen, L.Q.: Galerkin methods for natural frequencies of high-speed axially moving beams. J. Sound Vib. 329, 3484–3494 (2010)

    Article  Google Scholar 

  28. Ding, H., Chen, L.Q.: Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams. Acta Mech. Sin. 27, 426–437 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang, X.D., Chen, L.Q., Zu, J.W.: Vibrations and stability of an axially moving rectangular composite plate. J. Appl. Mech.-T ASME 78, 011018 (2011)

    Article  Google Scholar 

  30. Tang, Y.-Q., Chen, L.-Q.: Nonlinear free transverse vibrations of in-plane moving plates: without and with internal resonances. J. Sound Vib. 330, 110–126 (2011)

    Article  Google Scholar 

  31. Ghayesh, M.H., Kazemirad, S., Amabili, M.: Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech. Mach. Theory 52, 18–34 (2012)

    Article  Google Scholar 

  32. Zhang, Y.L., Chen, L.Q.: Internal resonance of pipes conveying fluid in the supercritical regime. Nonlinear Dyn. 67, 1505–1514 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Bishop, R.E.D., Johnson, D.C.: The Mechanics of Vibration. Cambridge University Press, Cambridge (1979)

    MATH  Google Scholar 

  34. Lang, G.F.: Matrix madness and complex confusion. A review of complex modes from multiple viewpoints. Sound Vib. 46, 8–12 (2012)

    Article  Google Scholar 

  35. Brake, M.R., Wickert, J.A.: Modal analysis of a continuous gyroscopic second-order system with nonlinear constraints. J. Sound Vib. 329, 893–911 (2010)

    Article  MATH  Google Scholar 

  36. Kerschen, G., Peeters, M., Golinval, J.C., Vakakis, A.F.: Nonlinear normal modes. Part I: a useful framework for the structural dynamicist. Mech. Syst. Signal Process. 23, 170–194 (2009)

    Article  Google Scholar 

  37. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes. Part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23, 195–216 (2009)

    Article  Google Scholar 

  38. Hill, T.L., Cammarano, A., Neild, S.A., Wagg, D.J.: Out-of-unison resonance in weakly nonlinear coupled oscillators. Proc. Math. Phys. Eng. Sci. R. Soc. 471, 20140659 (2015)

    Article  MathSciNet  Google Scholar 

  39. Neild, S.A., Champneys, A.R., Wagg, D.J., Hill, T.L., Cammarano, A.: The use of normal forms for analysing nonlinear mechanical vibrations. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 373, 20140404 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Shaw, S.W., Pierre, C.: Normal-modes of vibration for nonlinear continuous systems. J. Sound Vib. 169, 319–347 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Orloske, K., Leamy, M.J., Parker, R.G.: Flexural–torsional buckling of misaligned axially moving beams. I. Three-dimensional modeling, equilibria, and bifurcations. Inte. J. Solids Struct. 43, 4297–4322 (2006)

    Article  MATH  Google Scholar 

  43. Ghayesh, M.H., Kafiabad, H.A., Reid, T.: Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int. J. Solids Struct. 49, 227–243 (2012)

    Article  Google Scholar 

  44. Shahgholi, M., Khadem, S.E., Bab, S.: Free vibration analysis of a nonlinear slender rotating shaft with simply support conditions. Mech. Mach. Theory 82, 128–140 (2014)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Engineering, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    Xiao-Dong Yang, Ming Liu, Ying-Jing Qian, Song Yang & Wei Zhang

Authors
  1. Xiao-Dong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ming Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ying-Jing Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Song Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Dong Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, XD., Liu, M., Qian, YJ. et al. Linear and nonlinear modal analysis of the axially moving continua based on the invariant manifold method. Acta Mech 228, 465–474 (2017). https://doi.org/10.1007/s00707-016-1720-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1720-4

Search

Navigation