Multi-body dynamics and electromechanics: From a differential-geometric point of view

Abstract

Mechanics is the origin of physics. Almost any physical theory like electrodynamics stems from mechanical explanations. The mathematical-geometric considerations in mechanics serve as a prototype for other physical theories. Consequently, developments in modern physics in turn have a feedback to mechanics in terms of its representation. The laws of nature can be expressed as differential equations. The fact that these equations can be solved by average computers has led most engineers and many mathematical physicists to neglect geometrical aspects for solving and better understanding their problems. The intimate relation between geometry and analysis led to the differential geometry, which is a valuable tool for a better understanding in many physical disciplines like classical mechanics, electrodynamics, and nowadays in mechatronics. It has been the development of the theory of relativity that revealed the paramount importance of the differential geometry. Many problems in research and development can be studied by differential-geometric methods. Modern non-linear control theories, for instance, are entirely based on the differential geometry. This paper addresses some aspects in mathematical modelling of multi-body and electromechanical systems. The motivation for this research arises from applications of linear induction machines in modern transport technologies.

Keywords

differential geometry multi-body system dynamics Lagrangian mechanics Hamiltonian mechanics perturbed equations Riemannian geometry Finsler geometry

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References

Marsden

J. E.

Ratiu

T. S.

Introduction to Mechanics and Symmetry, 1994 (Springer-Verlag, New York, Berlin).

Arnold

V. J.

Mathematical Methods of Classical Mechanics, 1989 (Springer Verlag, Berlin).

Bullo

Lewis

A. D.

Geometric Control of Mechanical Systems. Modeling, Analysis and Design for Simple Mechanical Control Systems, 2005 (Springer Science and Business Media, New York).

Rasband

S. N.

Dynamics, 1983 (John Wiley and Sons, New York).

Schutz

Geometrical Methods of Mathematical Physics, 1980 (Cambridge University Press, London).

Agrawal

O. M.

Saigal

Dynamical analysis of multi-body systems using tangent coordinates

Comput. Structures, 1989, 31 (3), 349–355.

Blajer

Dynamic analysis of constrained multi-body systems in orthonormalized tangent space

Advanced Multibody System Dynamics (Ed. Schiehlen

) 1993, pp. 415–420 (Kluwer Academic Publishers, Dordrecht/Boston/London).

Chevallier

D. P.

Lie groups and multi-body dynamics. In Proceedings of the European Mechanics Colloquium EUROMECH 320 Czech Technical University, Prague, 6-8 June 1994, pp. 1–20.

Jungnickel

Differential-algebraic equations in Riemannian spaces and applications to multi-body system dynamics

Z. Angew. Math. Mech., 1994, 74 (9), 409–415.

10.

Maisser

Analytical dynamics of multi-body systems

Comput. Methods Appl. Mech. Engng, 1991, 91, 1391–1396.

11.

Maisser

A differential-geometric approach to the multi-body system dynamics

Z. Angew. Math. Mech., 1991, 71 (4), T116–T119.

12.

Maisser

Dynamik hybrider Mehrkörpersysteme aus kontinuumsmechanischer Sicht

Z. Angew. Math. Mech., 1996, 76 (1), 15–33.

13.

Kron

Non-Riemannian dynamics of rotating electrical machinery

J. Math. Phys., 1934, 13, 103–194.

14.

Rund

The Differential Geometry of Finsler Spaces, 1959 (Springer, Berlin).

15.

Maisser

Modellgleichungen für Manipulatoren

Tech. Mech., 1982, 3 (2), 64–78.

16.

Lovelock

Rund

Tensors, Differential Forms, and Variational Principles, 1975 (Dover, New York).

17.

Maisser

Analytische Dynamik von Mehrkörper-systemen

Z. Angew. Math. Mech., 1988, 68 (10), 463–481.

18.

Müller

Singuläre Phänomene in der Kinematik von Starrkörpermechanismen-mathematische Modellierung und lokale Analyse. PhD Thesis, Chemnitz University of Technology, 2004.

19.

Murray

R. M.

Sastry

S. S.

A Mathematical Introduction to Robotic Manipulation, 1993 (CRC Press, Boca Raton/Ann Arbor/London/Tokyo).

20.

Bloch

A. M.

Bailieul

Crouch

Marsden

Nonholonomic Mechanics and Control, 2003 (Springer-Verlag, New York).

21.

Steigenberger

Classical framework for non-holonomic mechanical control systems

Int. J. Robust Nonlinear Control, 1995, 5, 331–342.

22.

Hendel

Qualitative und quantitative Untersuchungen der Dynamik von Mehrkörpersystemen mittels Störungsgleichungen und 1. Integrale. Dissertation, Universität Leipzig, 1992.

23.

Kielau

Berechnung von Ljapunov-Exponenten für Mehrkörpersysteme

Z. Angew. Math. Mech., 1994, 74 (9), 436.

24.

Enge

Kielau

Maisser

Dynamiksimulation elektromechanischer Systeme. VDI-Fortschritt-Berichte, Reihe 20: Rechnergestützte Verfahren, Nr. 165, VDI-Verlag, Düsseldorf, 1995.

25.

Freudenberg

Kielau

Maisser

Simulation integrierter elektromechanischer Mehrkoordinatenantriebe. VDI-Fortschritt-Berichte, Reihe 20: Rechnergestützte Verfahren, Nr. 314, VDI-Verlag, Düsseldorf, 2000.

26.

Koenig

H. E.

Blackwell

W. A.

Electromechanical System Theory, 1961 (McGraw-Hill, New York).

27.

Maisser

Steigenberger

Lagrange-Formalismus für diskrete elektromechanische Systeme

Z. Angew. Math. Mech., 1979, 59, 717–730.

28.

Maisser

Ein Beitrag zur Theorie diskreter elektromechanischer Systeme mit Anwendungen in der Manipulator-Robotertechnik. B-Dissertation, Technische Universität Ilmenau, 1984.

29.

Maisser

Enge

Freudenberg

Kielau

Electromechanical interactions in multi-body systems containing electromechanical drives

Multibody Syst. Dyn., 1997, 1, 281–302.

30.

White

D. C.

Woodson

H. H.

Electromechanical Energy Conversion, 1961 (John Wiley and Sons, New York).

31.

Santilli

R. M.

The Foundation of Theoretical Mechanics, Part 1, The Inverse Problem in Newtonian Mechanics, 1978 (Springer-Verlag, New York/Heidelberg/Berlin).

32.

Helmholtz

Von Über die physikalische Bedeutung des Prinzips der kleinsten Wirkung

Wissenschaftliche Abhandlungen, 1885, III, 203–248.