Research Papers

An Ordinary Differential Equation Formulation for Multibody Dynamics: Holonomic Constraints

[+] 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 Computers and Information Division of ASME for publication in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscript received November 6, 2015; final manuscript received March 15, 2016; published online May 3, 2016. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 16(2), 021007 (May 03, 2016) (13 pages) Paper No: JCISE-15-1362; doi: 10.1115/1.4033237 History: Received November 06, 2015; Revised March 15, 2016

A method is presented for formulating and numerically integrating ordinary differential equations (ODEs) of motion for holonomically constrained multibody systems. Tangent space coordinates are defined as independent generalized coordinates that serve as state variables in the formulation, yielding ODEs of motion. Orthogonal dependent coordinates are used to enforce kinematic constraints at position, velocity, and acceleration levels. Criteria that assure accuracy of constraint satisfaction and well conditioning of the reduced mass matrix in the equations of motion are used as the basis for redefining local coordinates on the constraint manifold, as needed, transparent to the user and at minimal computational cost. The formulation is developed for holonomically constrained multibody models that are based on essentially any form of generalized coordinates. A spinning top with Euler parameter orientation coordinates is used as a model problem to analytically reduce Euler's equations of motion to ODEs. Numerical results using a fourth-order Nystrom integrator verify that accurate results are obtained, satisfying position, velocity, and acceleration constraints to computer precision. A computational algorithm for implementing the approach with state-of-the-art explicit numerical integrators is presented and used in solution of three examples, one planar and two spatial. Performance of the method in satisfying all three forms of kinematic constraint, based on error tolerances embedded in the formulation, is verified.

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Haug, E. J. , 1989, Computer-Aided Kinematics and Dynamics of Mechanical Systems, Allyn and Bacon, Boston, MA.
Pars, L. A. , 1979, A Treatise on Analytical Dynamics, Ox Bow Press, Woodbridge, CT.
Hairer, E. , Norsett, S. P. , and Wanner, G. , 1993, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd ed., Springer-Verlag, Berlin.
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]
Potra, F. A. , and Rheinboldt, W. C. , 1991, “ On the Numerical Solution of the Euler-Lagrange Equations,” Mech. Struct. Mach., 19(1), pp. 1–18. [CrossRef]
Petzold, L. R. , and Potra, F. A. , 1992, “ ODAE Methods for the Numerical Solution of Euler-Lagrange Equations,” Appl. Numer. Math., 10(5), pp. 397–413. [CrossRef]
Betsch, P. , 2005, “ The Discrete Null Space Method for the Energy Consistent Integration of Constrained Mechanical Systems; Part I: Holonomic Constraints,” Comput. Methods Appl. Mech. Eng., 194(50–52), pp. 5159–5190. [CrossRef]
Aghili, F. , 2005, “ A Unified Approach for Inverse and Direct Dynamics of Constrained Multibody Systems Based on Linear Projection Operator: Applications to Control and Simulation,” IEEE Trans. Rob., 21(5), pp. 834–849. [CrossRef]
Hairer, E. , 2011, Solving Differential Equations on Manifolds, Lecture Notes, Universite de Geneve, Geneva, Switzerland.
Muller, A. , and Terze, Z. , 2014, “ The Significance of the Configuration Space Lie Group for the Constraint Satisfaction in Numerical Time Integration of Multibody Systems,” Mech. Mach. Theory, 82, pp. 173–202. [CrossRef]
Eich-Soellner, E. , and Fuhrer, K. , 1998, Numerical Methods in Multibody Dynamics, B. G. Teubner, Stuttgart, Germany.
Coddington, E. A. , and Levinson, N. , 1955, Theory of Ordinary Differential Equations, McGraw-Hill, New York.
Maggi, G. A. , 1896, Principii della Teoria Matematica del Movimento dei Corpi: Corso de Meccanica Razionale, Ulrico Hoepli, Milano, Italy.
Singh, R. P. , and Likins, P. W. , 1985, “ Singular Value Decomposition for Constrained Dynamical Systems,” ASME J. Appl. Mech., 52(4), pp.943–948. [CrossRef]
Kim, S. S. , and Vanderploeg, M. J. , 1986, “ QR Decomposition for State Space Representation of Constrained Mechanical Dynamic Systems,” ASME J. Mech., Trans. Autom. Des., 108(2), pp. 183–188. [CrossRef]
Garcia de Jalon, J. , and Bayo, E. , 1994, Kinematic and Dynamic Simulation of Multibody Systems, Springer-Verlag, Berlin.
Borri, M. , Bottasso, C. , and Mantegazza, P. , 1992, “ Acceleration Projection Method in Multibody Dynamics,” Eur. J. Mech., A/Solids, 11(3), pp. 403–418.
Liang, C. G. , and Lance, G. M. , 1987, “ A Differentiable Null Space Method for Constrained Dynamic Analysis,” ASME J. Mech., Trans. Autom. Des., 109(3), pp. 405–411.
Wehage, R. A. , and Haug, E. J. , 1982, “ Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems,” ASME J. Mech. Des., 104(1), pp. 247–255. [CrossRef]
Negrut, D. , Haug, E. J. , and German, H. C. , 2003, “ An Implicit Runge–Kutta Method For Integration of Differential-Algebraic Equations of Multibody Dynamics,” Multibody Syst. Dyn., 9(2), pp. 121–142. [CrossRef]
Mani, N. K. , Haug, E. J. , and Atkinson, K. E. , “ Application of Singular Value Decomposition for Analysis of Mechanical System Dynamics,” ASME J. Mech., Trans. Autom. Des., 107(1), pp. 82–87. [CrossRef]
Arnold, V. I. , 1978, Mathematical Methods of Classical Mechanics, Springer, New York.
Serban, R. , and Haug, E. J. , 1998, “ Kinematic and Kinetic Derivatives in Multibody System Analysis,” Mech. Struct. Mach., 26(2), pp. 145–173. [CrossRef]


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Fig. 1

Heavy symmetric top with tip fixed

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Fig. 2

Orthogonal projection onto constraint manifold

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Fig. 3

x–y trajectory of centroid

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Fig. 4

Projection onto constraint manifold

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Fig. 5

Continuation of solution trajectory over charts

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Fig. 6

Planar double pendulum

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Fig. 7

Rotation ϕ2 of body 2 versus time

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Fig. 8

Top with tip constrained to x–y plane

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Fig. 9

x–y trajectories for tip and centroid, K = 0

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Fig. 10

x–y trajectories of tip and centroid, K = 50

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Fig. 11

Two-body spatial pendulum

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Fig. 12

x-coordinate of centroid of body 2 versus time

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Fig. 13

Constraint reaction force in the bar between points P1 and P2 versus time



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