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Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation

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Abstract

Finite element models of machine tools or their building blocks are usually very large and thus do not allow for fast simulation or application in controller design. Especially when algebraic constraints come into play the models become differential algebraic equations and therefore are even more difficult to handle in the application. In this contribution we propose a method based on modern system theoretic model order reduction algorithms that allows to generate a first order standard state space reduced order model for a structural model of an adaptive spindle support that is of second order index 1 differential algebraic form due to the piezo actuation applied. The accuracy of the method is demonstrated by a numerical frequency domain error analysis.

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References

  1. Antoulas AC (2005) Approximation of large-scale dynamical systems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  2. Benner P, Bonin T, Faßbender H, Saak J, Soppa A, Zaeh M (2009) Novel model reduction techniques for control of machine tools. In: Proceedings of ANSYS conference & 27. CADFEM users’ meeting 2009, 18–20 Nov 2009, Congress Center Leipzig. CADFEM GmbH, Grafing (2009). ISBN/ISSN: 3-937523-06-5

  3. Benner P, Kranz B, Saak J, Uddin MM (2012) Efficient reduced order state space model computation for a class of second order index one systems. Proc Appl Math Mech (submitted)

  4. Benner P, Kürschner P, Saak J (2012) Efficient handling of complex shift parameters in the low-rank Cholesky factor ADI method. Numer Alg pp 1–27. doi:10.1007/s11075-012-9569-7

  5. Benner P, Saak J (2011) Efficient balancing based MOR for large scale second order systems. Math Comput Model Dyn Syst 17(2):123–143. doi:10.1080/13873954.2010.540822

    Article  MathSciNet  MATH  Google Scholar 

  6. Bonin T, Zäh M, Soppa A, Faßbender H, Saak J, Benner P (2009) Modale versus moderne Ordnungsreduktionsverfahren: Effiziente Simulation von Werkzeugmaschinen. Mechatronik 117(11–12): 46–47. ISBN/ISSN:1867-2590

    Google Scholar 

  7. Brecher C, Manoharan D, Ladra U, Köpken HG (2010) Chatter suppression with an active workpiece holder. Prod Eng 4(2):239–245. doi:10.1007/s11740-009-0204-y

    Article  Google Scholar 

  8. Brecher C, Schauerte G, Merz M (2007) Modeling and simulation of adaptronic drilling tool axes as the basis of control design. Prod Eng 1:297–301. doi:10.1007/s11740-007-0049-1

    Article  Google Scholar 

  9. Drossel WG, Wittstock V (2008) Adaptive spindle support for improving machining operations. CIRP Ann Manuf Technol 57(1): 395–398. http://www.sciencedirect.com/science/article/pii/S0007850608000401

  10. Freitas F, Rommes J, Martins N (2008) Gramian-based reduction method applied to large sparse power system descriptor models. IEEE Trans Power Syst 23(3):1258–1270

    Article  Google Scholar 

  11. Kranz B (2009) Zustandsraumbeschreibung von piezo-mechanischen Systemen auf Grundlage einer Finite-Elemente-Diskretisierung. In: Analysis conference & 27th CADFEM users’ meeting 2009. Congress Center Leipzig, Germany

  12. Neugebauer R, Denkena B, Wegener K (2007) Mechatronic systems for machine tools. CIRP Ann Manuf Technol 56(2): 657–686. http://www.sciencedirect.com/science/article/pii/S0007850607001618

  13. Neugebauer R, Drossel WG, Bucht A, Kranz B, Pagel K (2010) Control design and experimental validation of an adaptive spindle support for enhanced cutting processes. CIRP Ann Manuf Technol 59(1): 373–376. http://www.sciencedirect.com/science/article/pii/S0007850610000302

  14. Neugebauer R, Drossel WG, Pagel K, Kranz B (2010) Making of state space models of piezo-mechanical systems with exact impedance mapping and strain output signals. In: Mechatronics 2010, 12th Mechatronics Forum Biennial International Conference vol 2. Eidgenössische Technische Hochschule Zürich, Zürich, pp 73–80

  15. Neugebauer R, Drossel WG, Ullrich M, Junker T (2011) Feinstkonturierung mittels mechatronischer Spindel: Betrachtung der Verfahrenssubstitution Formhonen durch Feinbohren. wt Werkstattstechnik online 101(1/2):47–53

    Google Scholar 

  16. Neugebauer R, Pagel K, Bucht A, Wittstock V, Pappe A (2010) Control concept for piezo-based actuator-sensor-units for uniaxial vibration damping in machine tools. Prod Eng 4:413–419. doi:10.1007/s11740-010-0237-2

    Article  Google Scholar 

  17. Neugebauer R, Wittstock V (2003) Parallelkinemematische experimentalplattform. Konstruktion 55(10):35–37

    Google Scholar 

  18. Penzl T (2000) Lyapack users guide. Technical report SFB393/00-33, Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern, TU Chemnitz, 09107 Chemnitz, Germany (2000). Available from http://www.tu-chemnitz.de/sfb393/sfb00pr.html

  19. Tisseur F, Meerbergen K (2001) The quadratic eigenvalue problem. SIAM Rev 43(2):235–286

    Article  MathSciNet  MATH  Google Scholar 

  20. Uddin M (2011) Model reduction for piezo-mechanical systems using balanced truncation. Master’s thesis, University of Stockholm. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-78227

  21. Zaeh M, Hennauer M (2011) Prediction of the dynamic behaviour of machine tools during the design process using mechatronic simulation models based on finite element analysis. Prod Eng 5(3):315–320. doi:10.1007/s11740-011-0303-4

    Article  Google Scholar 

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Acknowledgments

M. Monir Uddin would like to express his thanks to Prof. Boris Shapiro for the tutoring during his M.Sc. studies at the University of Stockholm. Some results contained in this paper were obtained by the first author in his Master thesis, supervised by the second and fourth authors at Chemnitz University of Technology.

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Authors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Mohammad Monir Uddin, Jens Saak & Peter Benner

  2. Department of Mathematics, Chemnitz University of Technology, 09107, Chemnitz, Germany

    Jens Saak & Peter Benner

  3. Fraunhofer Institute for Machine Tools and Forming Technology, Nöthnitzer Str. 44, 01187, Dresden, Germany

    Burkhard Kranz

Authors
  1. Mohammad Monir Uddin
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  2. Jens Saak
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  3. Burkhard Kranz
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  4. Peter Benner
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Correspondence to Jens Saak.

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Uddin, M.M., Saak, J., Kranz, B. et al. Computation of a compact state space model for an adaptive spindle head configuration with piezo actuators using balanced truncation. Prod. Eng. Res. Devel. 6, 577–586 (2012). https://doi.org/10.1007/s11740-012-0410-x

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  • DOI: https://doi.org/10.1007/s11740-012-0410-x

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