Skip to main content
SpringerLink
Log in

Reduced Order Modelling for Non-linear Rotating Systems in ALE Formulation with Contact

  • Conference paper
  • First Online:
Book cover Nonlinear Dynamics, Volume 1

Abstract

One approach for the simulation of rotating systems is the Arbitrary-Lagrangian-Eulerian (ALE) finite element formulation, which is well-established in the field of rolling contact mechanics for tires. With this formulation the rotational motion is handled from an Eulerian viewpoint and thus can be separated from the occurring Lagrangian deformation of the finite element mesh. In this context of (non-linear) systems undergoing gyroscopic and/or contact forces, e.g., for tires or disc brakes, model reduction techniques such as the Second order modal truncation, the Krylov subspace technique and the Craig-Bampton method are employed and analysed in their applicability.

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

Notes

  1. 1.

    This stands in agreement with the fact that the spectrum of eigenvalues and thus the eigenvectors for quadratic eigenvalues problems are (at least) symmetric about the real axis.

  2. 2.

    Inner/outer diameter di = 50 mm and da = 300 mm, thickness t = 10 mm, Young’s modulus E = 2.1 ⋅ 105 MPa, Poisson ratio ν = 0.3.

  3. 3.

    α = 0, β = 1 ⋅ 10−6.

  4. 4.

    Alternatively bi-modal decoupling, using the left and right eigenvectors and their transposed forms, or working in complex coordinates for decoupling is possible, [1] and [16].

References

  1. Gasch, R., Knothe, K., Liebich, R.: Strukturdynamik: Diskrete Systeme und Kontinua. Springer, Berlin (2012)

    Google Scholar 

  2. Ewins, D.J.: Modal Testing: Theory, Practice and Application, 2nd edn. Research Studies Press, Baldock/Philadelphia (2000); Previous edition (1995)

    Google Scholar 

  3. Salimbahrami, B.: Structure preserving order reduction of large scale second order models. PhD thesis, Lehrstuhl für Regelungstechnik, Technische Universität München (2005)

    Google Scholar 

  4. Villemagne, C.D., Skelton, R.E.: Model reductions using a projection formulation. Int. J. Control. 46(6), 2141–2169 (1987)

    Article  MathSciNet  Google Scholar 

  5. Freund, R.W.: Krylov-subspace methods for reduced-order modeling in circuit simulation. J. Comput. Appl. Math. 123(1), 395–421 (2000)

    Article  MathSciNet  Google Scholar 

  6. Arnoldi, W.E.: The principle of minimized iterations in the solution of the matrix eigenvalue problem. Q. Appl. Math. 9(1), 17–29 (1951)

    Article  MathSciNet  Google Scholar 

  7. Lanczos, C.: An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Natl. Bur. Stand. B 45, 255–282 (1950)

    Article  MathSciNet  Google Scholar 

  8. Lohmann, B., Salimbahrami, B.: Reduction of second order systems using second order Krylov subspaces. IFAC Proc. Vol. 38(1), 614–619 (2005). 16th IFAC World Congress

    Article  Google Scholar 

  9. Salimbahrami, B., Lohmann, B.: Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl. 415(2), 385–405 (2006). Special Issue on Order Reduction of Large-Scale Systems

    Article  MathSciNet  Google Scholar 

  10. Bampton, M.C.C., Craig, R.R., Jr.: Coupling of substructures for dynamic analyses. AIAA J. 6, 1313–1319 (1968)

    Article  Google Scholar 

  11. Nackenhorst, U.: The ALE-formulation of bodies in rolling contact: theoretical foundations and finite element approach. Comput. Methods Appl. Mech. Eng. 193(39), 4299–4322 (2004)

    Article  MathSciNet  Google Scholar 

  12. Nackenhorst, U.: Rollkontaktdynamik, Numerische Analyse der Dynamik rollender Körper mit der Finite Elemente Methode. Habilitation, Institut für Mechanik, Universität der Bundeswehr Hamburg (2000)

    Google Scholar 

  13. Willner, K.: Kontinuums- und Kontaktmechanik: Synthetische und Analytische Darstellung. Springer, Berlin/Heidelberg (2003)

    Chapter  Google Scholar 

  14. Geisler, J., Willner, K.: Modeling of jointed structures using zero thickness interface elements. PAMM 7(1), 4050009–4050010 (2007)

    Article  Google Scholar 

  15. Ziefle, M., Nackenhorst, U.: Numerical techniques for rolling rubber wheels: treatment of inelastic material properties and frictional contact. Comput. Mech. 42(3), 337–356 (2008)

    Article  MathSciNet  Google Scholar 

  16. Genta, G.: Dynamics of Rotating Systems. Mechanical Engineering Series. Springer, New York (2005)

    Google Scholar 

  17. Bryan, G.H.: On the beats in the vibrations of a revolving cylinder or bell. Proc. Camb. Philos. Soc. 7(3), 101–114 (1890)

    Google Scholar 

  18. Joubert, S.V., Fedotov, I., Pretorius, W., Shatalov, M.: On gyroscopic effects in vibrating and axially rotating solid and annular discs. In: 2007 International Conference – Days on Diffraction, pp. 89–94 (2007)

    Google Scholar 

  19. Farhat, C., Avery, P., Chapman, T., Cortial, J.: Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int. J. Numer. Methods Eng. 98(9), 625–662 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Friedrich-Alexander Universität Erlangen-Nürnberg, Erlangen, Germany

    Tim Weidauer & Kai Willner

Authors
  1. Tim Weidauer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kai Willner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tim Weidauer .

Editor information

Editors and Affiliations

  1. Mechanical Engineering, S3L, Quartier Polytech, University of Liege, Department of Aerospace, Liege, Belgium

    Gaetan Kerschen

Rights and permissions

Reprints and permissions

Copyright information

© 2019 The Society for Experimental Mechanics, Inc.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Weidauer, T., Willner, K. (2019). Reduced Order Modelling for Non-linear Rotating Systems in ALE Formulation with Contact. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-74280-9_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-74280-9_31

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-74279-3

  • Online ISBN: 978-3-319-74280-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation