Abstract
This work considers a class of multibody dynamic systems involving bilateral nonholonomic constraints. An appropriate set of equations of motion is employed first. This set is derived by application of Newton’s second law and appears as a coupled system of strongly nonlinear second-order ordinary differential equations in both the generalized coordinates and the Lagrange multipliers associated with the motion constraints. Next, these equations are manipulated properly and converted to a weak form. Furthermore, the position, velocity and momentum type quantities are subsequently treated as independent. This yields a three-field set of equations of motion, which is then used as a basis for performing a suitable temporal discretization, leading to a complete time integration scheme. In order to test and validate its accuracy and numerical efficiency, this scheme is applied next to challenging mechanical examples, exhibiting rich dynamics. In all cases, the emphasis is put on highlighting the advantages of the new method by direct comparison with existing analytical solutions as well as with results of current state-of-the-art numerical methods. Finally, a comparison is also performed with results available for a benchmark problem.
Similar content being viewed by others
References
Geradin, Μ, Cardona, Α: Flexible Multibody Dynamics. Wiley , NY (2001)
Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, NY (2005)
Bauchau, O.A.: Flexible Multibody Dynamics. Springer, London (2011)
Bloch, A.M.: Nonholonomic Mechanics and Control. Springer, NY (2003)
Νeimark, J.I., Fufaev, N.A.: Dynamics of Nonholonomic Systems. Translations of Mathematical Monographs, American Mathematical Society 33, Providence, RI (1972)
Βrenan, K.E., Campbell, S.L., Petzhold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, ΝY (1989)
Bauchau, ΟΑ, Epple, Α, Bottasso, C.L.: Scaling of constraints and augmented Lagrangian formulations in multibody dynamics simulations. ASME J. Comput. Nonlinear Dyn. 4, 021007 (2009)
García Orden, J.C.: Energy considerations for the stabilization of constrained mechanical systems with velocity projection. Nonlinear Dyn. 60, 49–62 (2010)
García Orden, J.C., Conde, S.C.: Controllable velocity projection for constraint stabilization in multibody dynamics. Nonlinear Dyn. 68, 245–257 (2012)
Paraskevopoulos, E., Natsiavas, S.: On application of Newton’s law to mechanical systems with motion constraints. Nonlinear Dyn. 72, 455–475 (2013)
Natsiavas, S., Paraskevopoulos, E.: A set of ordinary differential equations of motion for constrained mechanical systems. Nonlinear Dyn. 79, 1911–1938 (2015)
Paraskevopoulos, E., Natsiavas, S.: Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems. Int. J. Non-linear Mech. 77, 208–222 (2015)
Washizu, K.: Variational Methods in Elasticity and Plasticity, 3rd edn. Pergamon Press, Oxford (1982)
Felippa, C.: On the original publication of the general canonical functional of linear elasticity. J. Appl. Mech. 67, 217–219 (2000)
Potosakis, N., Paraskevopoulos, E., Natsiavas, S.: Application of an augmented Lagrangian approach to multibody systems with equality motion constraints. Nonlinear Dyn. 99, 753–776 (2020)
Greenwood, D.T.: Principles of Dynamics. Prentice-Hall Inc., Englewood Cliffs, NJ (1988)
Murray, R.M., Li, Ζ, Sastry, S.S.: A Mathematical Introduction to Robot Manipulation. CRC Press, Boca Raton, FL (1994)
Papastavridis, J.G.: Tensor Calculus and Analytical Dynamics. CRC Press, Boca Raton (1999)
Frankel, T.: The Geometry of Physics: An Introduction. Cambridge University Press, NY (1997)
Rektorys, K.: Variational Methods in Mathematics Science and Engineering. D. Reidel Publishing Company, Dordrecht (1977)
Ženišek, A.: Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. Academic Press, London (1990)
Paraskevopoulos, E., Natsiavas, S.: A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory. Int. J. Solids Struct. 50, 57–72 (2013)
Paraskevopoulos, E.A., Panagiotopoulos, C.G., Talaslidis, D.G.: Rational derivation of energy/momentum-preserving time integration algorithms. In: Application to dynamic response under moving vehicles. ECCOMAS Thematic Conf. Comput. Meth. Struct. Dyn. Earthq. Eng. Crete, Greece (2007)
Meijaard, J.P., Papadopoulos, J.M., Ruina, A., Schwab, A.L.: Linearized dynamics equations for the balance and steer of a bicycle: a benchmark and review. Proc. R. Soc. A Math. Phys. Eng. Sci. 463 (2084), 1955–1982 (2007)
García-Agúndez, A., García-Vallejo, D., Freire, E.: Linearization approaches for general multibody systems validated through stability analysis of a benchmark bicycle model. Nonlinear Dyn. 103, 557–580 (2021)
MSC ADAMS 2018.1, User Guide, MSC Software Corporation, California, USA.
MotionSolve v19.0, User Guide, Altair Engineering Inc., Irvine, California, USA.
Bayo, E., Ledesma, R.: Augmented Lagrangian and mass-orthogonal projection methods for constrained multibody dynamics. Nonlinear Dyn. 9, 113–130 (1996)
Dopico, D., Gonzalez, F., Cuadrado, J., Kövecses, J.: Determination of holonomic and nonholonomic constraint reactions in an index-3 augmented Lagrangian formulation with velocity and acceleration projections. ASME J. Comput. Nonlinear Dyn. 9, p. 041006 (2014)
Gonzalez, F., Dopico, D., Pastorino, R., Cuadrado, J.: Behaviour of augmented Lagrangian and Hamiltonian methods for multibody dynamics in the proximity of singular configurations. Nonlinear Dyn. 85, 1491–1508 (2016)
IFToMM T.C. for Multibody Dynamics, Library of Computational Benchmark Problems, http://www.iftomm-multibody.org/benchmark
Papalukopoulos, C., Natsiavas, S.: Dynamics of large scale mechanical models using multi-level substructuring. ASME J. Comput. Nonlinear Dyn. 2, 40–51 (2007)
Lubarda, V.A.: Dynamics of a light hoop with an attached heavy disk: inside an interaction pulse. J. Mech. Mat. Struct. 4, 1027–1040 (2009)
Bronars, A., O’Reilly, O.M.: Gliding motions of a rigid body: the curious dynamics of Littlewood’s rolling hoop. Proc. R. Soc. A 475, 20190440 (2019)
Antali, M., Stepan, G.: Nonsmooth analysis of three-dimensional slipping and rolling in the presence of dry friction. Nonlinear Dyn. 97, 1799–1817 (2019)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Passas, P., Natsiavas, S. & Paraskevopoulos, E. Numerical integration of multibody dynamic systems involving nonholonomic equality constraints. Nonlinear Dyn 105, 1191–1211 (2021). https://doi.org/10.1007/s11071-021-06500-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-021-06500-5