Skip to main content
SpringerLink
Log in

Simulation process of flexible multibody systems with non-modal model order reduction techniques

  • Published:
Multibody System Dynamics Aims and scope Submit manuscript

Abstract

One important issue for the simulation of flexible multibody systems is the reduction of the flexible body’s degrees of freedom. In this work, nonmodal model reduction techniques for flexible multibody systems within the floating frame of reference framework are considered. While traditionally in the multibody community modal techniques in many different forms are used, here other methods from system dynamics and mathematics are in the focus. For the reduction process, finite element data and user inputs are necessary. Prior to the reduction process, the user first needs to choose boundary conditions fitting the chosen reference frame before defining the appropriate in- and outputs. In this work, four different possibilities of modeling appropriate interface points to reduce the number of inputs and outputs are presented.

The main model reduction techniques to be considered in this context are moment-matching by projection on Krylov-subspaces, singular vaule decomposition (SVD)-based reduction techniques and combinations of those which are also compared to traditional modal approaches. All these reduction techniques are implemented in the model order reduction code Morembs. In addition, an error estimator for Krylov-subspace methods exists and an a-priori error bound can be calculated if frequency weighted Gramian matrices are used for the reduction process. This allows a fully automated reduction process. We evaluate and compare these methods in the frequency as well as in the time domain by reducing the flexible degrees of freedom of a rack used for active vibration damping of a scanning tunneling microscope.

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. Schwertassek, R., Wallrapp, O.: Dynamik flexibler Mehrkörpersysteme. Vieweg, Braunschweig (1999) (in German)

    Google Scholar 

  2. Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  3. Wu, S.-C., Haug, E.J., Kim, S.-S.: A variational approach to dynamics of flexible multibody systems. Mech. Struct. Mach., 17, 3–32 (1989)

    Article  MathSciNet  Google Scholar 

  4. Cardona, A., Géradin, M.: A superelement formulation for mechanism analysis. Comput. Methods Appl. Mech. Eng. 100, 1–29 (1992)

    Article  MATH  Google Scholar 

  5. Schwertassek, R., Wallrapp, O., Shabana, A.: Flexible multibody simulation and choice of shape functions. Nonlinear Dyn. 20, 361–380 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Wallrapp, O., Wiedemann, S.: Comparison of results in flexible multibody dynamics using various approaches. Nonlinear Dyn. 34, 189–206 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  8. Wallrapp, O.: Standardization of flexible body modeling in multibody system codes, Part I: Definition of standard input data. Mech. Struct. Mach. 22, 283–304 (1994)

    Article  Google Scholar 

  9. Lehner, M., Eberhard, P.: A two-step approach for model reduction in flexible multibody dynamics. Multibody Syst. Dyn. 17, 157–176 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fehr, J., Eberhard, P., Lehner, M.: Improving the reduction process in flexible multibody dynamics by the use of 2nd order position Gramian matrices. In: Proceedings 6th EUROMECH Nonlinear Dynamics Conference, St. Petersburg, Russia (2008)

    Google Scholar 

  11. Lu, J., Ast, A., Eberhard, P.: Modeling and active vibration control of flexible structures using multibody system theory. In: Proceedings of ICMEM, Wuxi, China (2007)

    Google Scholar 

  12. Nikravesh, P.E., Lin, Y.-S.: Use of principal axes as the floating reference frame for a moving deformable body. Multibody Syst. Dyn. 13, 211–231 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Heckmann, A.: On the choice of boundary conditions for mode shapes in flexible multibody systems. Multibody Syst. Dyn. 23, 141–163 (2010)

    Article  MATH  Google Scholar 

  14. Release 11.0 Documentation for Ansys. http://www.ansys.com (2009)

  15. Dresig, H., Holzweißig, F.: Maschinendynamik, vol. 5. Springer, Berlin (2004) (in German)

    Google Scholar 

  16. Bavely, C.A., Stewart, G.W.: An algorithm for computing reducing subspaces by block diagonalization. SIAM J. Numer. Anal. 16, 359–367 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lehner, M.: Modellreduktion in elastischen Mehrkörpersystemen. In: Dissertation, Schriften aus dem Institut für Technische und Numerische Mechanik der Universität Stuttgart, vol. 10. Shaker, Aachen (2007) (in German)

    Google Scholar 

  18. MSC.Software Corporation: Theoretical Background of ADAMS/Flex, Part of the ADAMS/Flex Training Course, 2003rd ed. MSC Software, Santa Ana (2003)

    Google Scholar 

  19. Antoulas, T.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia (2005)

    Book  MATH  Google Scholar 

  20. Salimbahrami, B., Lohmann, B.: Order reduction of large scale second-order systems using Krylov subspace methods. Linear Algebra Appl. 415, 385–405 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Stykel, T.: Balanced truncation model reduction of second-order systems. In: Troch, I., Breitenecker, F. (eds.) 5th Vienna Symposium on Mathematical Modelling, Vienna (2006)

    Google Scholar 

  22. Bai, Z.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43, 9–44 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Beattie, C.A., Gugercin, S.: Krylov-based model reduction of second-order systems with proportional damping. In: 44th IEEE Conference on Decision and Control, 2005 European Control Conference, pp. 2278–2283 (2005)

    Chapter  Google Scholar 

  24. Han, J.S., Rudnyi, E.B., Korvink, J.: Efficient optimization of transient dynamic problems in MEMS devices using model order reduction. J. Micromech. Microeng. 15, 822–832 (2005)

    Article  Google Scholar 

  25. Gawronski, W.K.: Advanced Structural Dynamics and Active Control of Structures. Springer, New York (2004)

    Book  Google Scholar 

  26. Salimbahrami, S.B.: Structure preserving order reduction of large scale second order models. Ph.D. Thesis, Technische Universität München (2005)

  27. Bai, Z., Slone, R., Smith, W., Ye, Q.: Error bound for reduced system model by Pade approximation via the Lanczos process. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 18, 133–141 (1999)

    Article  Google Scholar 

  28. Gugercin, S., Antoulas, A.C., Beattie, C.A.: \(\mathcal{H}_{2}\) model reduction for large-scale linear dynamical systems. SIAM J. Matrix Anal. Appl. 30, 609–638 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Eid, R., Salimbahrami, B., Lohmann, B.: Equivalence of Laguerre-based model order reduction and moment matching. IEEE Trans. Autom. Control 52, 1104–1108 (2007)

    Article  MathSciNet  Google Scholar 

  30. Eid, R.: Time Domain Model Reduction by Moment Matching. Dr. Hut, München (2009)

    Google Scholar 

  31. Chu, C.-C., Lai, M.-H., Feng, W.-S.: MIMO Interconnects order reductions by using the multiple point adaptive-order rational global Arnoldi algorithm. IEICE Trans. Electron. E89-C, 792–802 (2006)

    Article  Google Scholar 

  32. Fehr, J., Eberhard, P.: Error-controlled model reduction in flexible multibody dynamics. J. Comput. Nonlinear Dyn. 5, 031005 (2010)

    Article  Google Scholar 

  33. Heyouni, M., Jbilou, K.: Matrix Krylov subspace methods for large scale model reduction problems. Appl. Math. Comput. 181, 1215–1228 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Grimme, E.J.: Krylov projection methods for model reduction. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1997)

  35. Konkel, Y., Farle, O., Dyczij-Edlinger, R.: Ein Fehlerschätzer für die Krylov-Unterraum basierte Ordnungsreduktion zeitharmonischer Anregungsprobleme. In: Lohmann, B., Kugi, A. (eds.) Tagungsband GMA-Fachausschuss 1.30 “Modellbildung, Identifikation und Simulation in der Automatisierungstechnik”, VDI/VDE-GMA, Technische Universität Wien, Institut für Automatisierungs- und Regelungstechnik, Salzburg (2008) (in German)

    Google Scholar 

  36. Laub, A.J., Laub, A.J., Heath, M.T., Paige, C.C., Ward, R.C.: Computation of system balancing transformations. In: 25th IEEE Conference on Decision and Control, Athens, Greece, pp. 548–553 (1986)

    Chapter  Google Scholar 

  37. Enns, D.: Model reduction with balanced realizations: an error bound and a frequency weighted generalization. In: Proceedings of the 23rd IEEE Conference on Decision and Control, Las Vegas, 1984, pp. 127–132. IEEE, New York (1984)

    Chapter  Google Scholar 

  38. Chahlaoui, Y., Lemonnier, D., Vandendorpe, A., Dooren, P.V.: Second-order balanced truncation. Linear Algebra Appl. 415, 378–384 (2006)

    Article  Google Scholar 

  39. Reis, T., Stykel, T.: Balanced truncation model reduction of second-order systems. Math. Comput. Model. Dyn. Syst. 14, 391–406 (2008)

    MATH  MathSciNet  Google Scholar 

  40. Benner, P., Saak, J.: Efficient balancing based MOR for second order systems arising in control of machine tools. In: Troch, I., Breitenecker, F. (eds.) Proceedings of the MathMod 2009 (2009)

    Google Scholar 

  41. Kruszinski, L.: Implementierung und Test neuer Methoden zur Approximation des dominanten Eigenraums von Gramschen Matrizen 2ter Ordnung. Thesis DIPL-130, Institute of Engineering and Computational Mechanics, University of Stuttgart (2008) (in German)

  42. Gasch, R., Knothe, K.: Strukturdynamik Band 2—Kontinua und ihre Diskretisierung (in German). Springer, Berlin (1989)

    Google Scholar 

  43. Gawronski, W.K.: Dynamics and Control of Structures—a Modal Approach. Springer, Berlin (1998)

    MATH  Google Scholar 

  44. Dietz, S.: Vibration and Fatigue Analysis of Vehicle Systems Using Component Modes. Fortschritt-Berichte VDI, reihe 12(401). VDI, Düsseldorf (1999)

    Google Scholar 

  45. Graig, R.R.J.: Structural Dynamics. Wiley, New York (1981)

    Google Scholar 

  46. Koutsovasilis, P., Beitelschmidt, M.: Comparison of model reduction techniques for large mechanical systems. Multibody Syst. Dyn. 20, 111–128 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  47. Dickens, J.M., Nakagawa, J.M., Wittbrodt, M.J.: A critique of mode acceleration and modal truncation augmentation methods for modal response analysis. Comput. Struct. 62, 985–998 (1997)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569, Stuttgart, Germany

    Jörg Fehr & Peter Eberhard

Authors
  1. Jörg Fehr
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peter Eberhard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Eberhard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fehr, J., Eberhard, P. Simulation process of flexible multibody systems with non-modal model order reduction techniques. Multibody Syst Dyn 25, 313–334 (2011). https://doi.org/10.1007/s11044-010-9238-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11044-010-9238-3

Keywords

Search

Navigation