Research Papers

Extension of Maggi and Kane Equations to Holonomic Dynamic Systems

[+] Author and Article Information
Edward J. Haug

Carver Distinguished Professor Emeritus
Department of Mechanical Engineering,
The University of Iowa,
Iowa City, IA 52242
e-mail: echaug@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 18, 2018; final manuscript received September 16, 2018; published online October 15, 2018. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 13(12), 121003 (Oct 15, 2018) (6 pages) Paper No: CND-18-1275; doi: 10.1115/1.4041579 History: Received June 18, 2018; Revised September 16, 2018

The Maggi and Kane equations of motion are valid for systems with only nonholonomic constraints, but may fail when applied to systems with holonomic constraints. A tangent space ordinary differential equation (ODE) extension of the Maggi and Kane formulations that enforces holonomic constraints is presented and shown to be theoretically sound and computationally effective. Numerical examples are presented that demonstrate the extended formulation leads to solutions that satisfy position, velocity, and acceleration constraints for holonomic systems to near computer precision.

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Maggi, G. A. , 1896, Principii Della Teoria Matematica Del Movimento Dei Corpi: Corso de Meccanica Razionale, Ulrico Hoepli, Milano.
Maggi, G. A. , 1901, “ Di Alcune Nuove Forme Delle Equazioni Della Dinamica Applicabili ai Sistemi Anolonomi,” Rendiconti Della Regia Academia Dei Lincei, Serie V, Vol. X, pp. 287–291.
Pars, L. A. , 1965, A Treatise on Analytical Dynamics, Reprint by Ox Bow Press (1979), Woodbridge, CT.
Rabier, P. J. , and Rheinboldt, W. C. , 2002, “ Theoretical and Numerical Analysis of Differential-Algebraic Equations,” Handbook of Numerical Analysis, Vol. 8, P. G. Ciarlet and J. L. Lions , eds., Elsevier Science B.V, Amsterdam, The Netherlands, pp. 183–540.
Teschl, G. , 2012, Ordinary Differential Equations and Dynamical Systems, American Math Society, Providence, RI.
Neimark, J. I. , and Fufaev, N. A. , 1972, Dynamics of Nonholonomic Systems, American Mathematical Society, Providence, RI.
Kane, T. R. , and Levinson, D. A. , 1985, Dynamics: Theory and Applications, McGraw-Hill, New York.
Borri, M. , Bottasso, C. , and Mantegazza, P. , 1990, “ Equivalence of Kane's and Maggi's Equations,” Meccanica, 25(4), pp. 272–274. [CrossRef]
Tseng, F.-C. , Ma, Z.-D. , and Hulbert, G. M. , 2003, “ Efficient Numerical Solution of Constrained Multibody Dynamics Systems,” Comput. Methods Appl. Mech. Eng., 192(3–4), pp. 439–472. [CrossRef]
Bauchau, O. A. , and Laulusa, A. , 2008, “ Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011005. [CrossRef]
Laulusa, A. , and Bauchau, O. A. , 2008, “ Review of Classical Approaches for Constraint Enforcement in Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011004. [CrossRef]
Garcia de Jalon, J. , Callejo, A. , and Hidalgo, A. F. , 2012, “ Efficient Solution of Maggi's Equations,” ASME J. Comput. Nonlinear Dyn., 7(2), p. 021003. [CrossRef]
Papastavridis, J. G. , 1990, “ The Maggi or Canonical Form of Lagrange's Equations of Motion of Holonomic Mechanical Systems,” ASME J. Appl. Mech., 57(4), pp. 1004–1010. [CrossRef]
Haug, E. J. , 2016, “ An Ordinary Differential Equation Formulation for Multibody Dynamics: Holonomic Constraints,” ASME J. Comput. Inf. Sci. Eng., 16(2), p. 021007. [CrossRef]
Strang, G. , 1980, Liner Algebra and Its Applications, 2nd ed., Academic Press, New York.
Corwin, L. J. , and Szczarba, R. H. , 1982, Multivariable Calculus, Marcel Dekker, New York.
Arnold, V. I. , 1978, Mathematical Methods of Classical Mechanics, Springer, New York.
Stuelpnagel, J. , 1964, “ On the Parametrization of the Three-Dimensional Rotation Group,” SIAM Rev., 6(4), pp. 422–430. [CrossRef]


Grahic Jump Location
Fig. 1

Projection onto constraint manifold

Grahic Jump Location
Fig. 2

Continuation of solution trajectory over charts

Grahic Jump Location
Fig. 3

Heavy symmetric top with tip constrained to xy plane

Grahic Jump Location
Fig. 4

Two body spatial pendulum



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