Research Papers

Assessment of Linearization Approaches for Multibody Dynamics Formulations

[+] Author and Article Information
Francisco González

Laboratorio de Ingeniería Mecánica,
Universidade da Coruña,
Ferrol 15403, Spain
e-mail: f.gonzalez@udc.es

Pierangelo Masarati

Dipartimento di Scienze
e Tecnologie Aerospaziali,
Politecnico di Milano,
Milano 20156, Italy
e-mail: pierangelo.masarati@polimi.it

Javier Cuadrado

Laboratorio de Ingeniería Mecánica,
Universidade da Coruña,
Ferrol 15403, Spain
e-mail: javicuad@cdf.udc.es

Miguel A. Naya

Associate Professor
Laboratorio de Ingeniería Mecánica,
Universidade da Coruña,
Ferrol 15403, Spain
e-mail: minaya@udc.es

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 13, 2016; final manuscript received November 25, 2016; published online January 20, 2017. Assoc. Editor: Paramsothy Jayakumar.

J. Comput. Nonlinear Dynam 12(4), 041009 (Jan 20, 2017) (7 pages) Paper No: CND-16-1493; doi: 10.1115/1.4035410 History: Received October 13, 2016; Revised November 25, 2016

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

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Bauchau, O. A. , 2011, Flexible Multibody Dynamics, Springer, Dordrecht, The Netherlands.
Mariti, L. , Belfiore, N. P. , Pennestrì, E. , and Valentini, P. P. , 2011, “ Comparison of Solution Strategies for Multibody Dynamics Equations,” Int. J. Numer. Methods Eng., 88(7), pp. 637–656. [CrossRef]
Marques, F. , Souto, A. P. , and Flores, P. , 2016, “ On the Constraints Violation in Forward Dynamics of Multibody Systems,” Multibody Syst. Dyn. (Online).
Ogata, K. , 2001, Modern Control Engineering, 4th ed., Prentice Hall, Upper Saddle River, NJ.
Grewal, M. S. , and Andrews, A. P. , 2015, Kalman Filtering: Theory and Practice With MATLAB, 4th ed., Wiley-IEEE Press, Hoboken, NJ.
Ripepi, M. , and Masarati, P. , 2011, “ Reduced Order Models Using Generalized Eigenanalysis,” Proc. Inst. Mech. Eng., Part K, 225(1), pp. 52–65.
Escalona, J. L. , and Chamorro, R. , 2008, “ Stability Analysis of Vehicles on Circular Motions Using Multibody Dynamics,” Nonlinear Dyn., 53(3), pp. 237–250. [CrossRef]
Masarati, P. , 2013, “ Estimation of Lyapunov Exponents From Multibody Dynamics in Differential-Algebraic Form,” Proc. Inst. Mech. Eng., Part K, 227(1), pp. 23–33.
Masarati, P. , and Tamer, A. , 2015, “ Sensitivity of Trajectory Stability Estimated by Lyapunov Characteristic Exponents,” Aerosp. Sci. Technol., 47, pp. 501–510. [CrossRef]
Masarati, P. , 2009, “ Direct Eigenanalysis of Constrained System Dynamics,” Proc. Inst. Mech. Eng., Part K, 223(4), pp. 335–342.
Gontier, C. , and Li, Y. , 1995, “ Lagrangian Formulation and Linearization of Multibody System Equations,” Comput. Struct., 57(2), pp. 317–331. [CrossRef]
Peterson, D. L. , Gede, G. , and Hubbard, M. , 2015, “ Symbolic Linearization of Equations of Motion of Constrained Multibody Systems,” Multibody Syst. Dyn., 33(2), pp. 143–161. [CrossRef]
Negrut, D. , and Ortiz, J. L. , 2006, “ A Practical Approach for the Linearization of the Constrained Multibody Dynamics Equations,” ASME J. Comput. Nonlinear Dyn., 1(3), pp. 230–239. [CrossRef]
González, F. , Masarati, P. , and Cuadrado, J. , 2016, “ On the Linearization of Multibody Dynamics Formulations,” ASME Paper No. DETC2016-59227.
Serna, M. A. , Avilés, R. , and García de Jalón, J. , 1982, “ Dynamic Analysis of Plane Mechanisms With Lower Pairs in Basic Coordinates,” Mech. Mach. Theory, 17(6), pp. 397–403. [CrossRef]
Jain, A. , 2011, “ Graph Theoretic Foundations of Multibody Dynamics. Part II: Analysis and Algorithms,” Multibody Syst. Dyn., 26(3), pp. 335–365. [CrossRef] [PubMed]
Jain, A. , 2012, “ Multibody Graph Transformations and Analysis—Part II: Closed-Chain Constraint Embedding,” Nonlinear Dyn., 67(3), pp. 2153–2170. [CrossRef] [PubMed]
Rong, B. , Rui, X. , and Wang, G. , 2010, “ Modified Finite Element Transfer Matrix Method for Eigenvalue Problem of Flexible Structures,” ASME J. Appl. Mech., 78(2), p. 021016. [CrossRef]
Baumgarte, J. , 1972, “ Stabilization of Constraints and Integrals of Motion in Dynamical Systems,” Comput. Methods Appl. Mech. Eng., 1(1), pp. 1–16. [CrossRef]
Bayo, E. , García de Jalón, J. , and Serna, M. A. , 1988, “ A Modified Lagrangian Formulation for the Dynamic Analysis of Constrained Mechanical Systems,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 183–195. [CrossRef]
Masarati, P. , 2011, “ Adding Kinematic Constraints to Purely Differential Dynamics,” Comput. Mech., 47(2), pp. 187–203. [CrossRef]
García de Jalón, J. , and Bayo, E. , 1994, Kinematic and Dynamic Simulation of Multibody Systems. The Real-Time Challenge, Springer-Verlag, New York.
Kamman, J. W. , and Huston, R. L. , 1984, “ Dynamics of Constrained Multibody Systems,” ASME J. Appl. Mech., 51(4), pp. 899–903. [CrossRef]
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]
Mani, N. K. , Haug, E. J. , and Atkinson, K. E. , 1985, “ Application of Singular Value Decomposition for Analysis of Mechanical System Dynamics,” J. Mech., Transm., Autom. Des., 107(1), pp. 82–87. [CrossRef]
Kim, S. S. , and Vanderploeg, M. J. , 1986, “ QR Decomposition for State Space Representation of Constrained Mechanical Dynamic Systems,” J. Mech., Transm., Autom. Des., 108(2), pp. 183–188. [CrossRef]
Liang, C. G. , and Lance, G. M. , 1987, “ A Differentiable Null Space Method for Constrained Dynamic Analysis,” J. Mech., Transm., Autom. Des., 109(3), pp. 405–411. [CrossRef]
Agrawal, O. P. , and Saigal, S. , 1989, “ Dynamic Analysis of Multi-Body Systems Using Tangent Coordinates,” Comput. Struct., 31(3), pp. 349–355. [CrossRef]
Dopico, D. , Zhu, Y. , Sandu, A. , and Sandu, C. , 2014, “ Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations,” ASME J. Comput. Nonlinear Dyn., 10(1), p. 011012. [CrossRef]
Moler, C. B. , and Stewart, G. W. , 1973, “ An Algorithm for Generalized Matrix Eigenvalue Problems,” SIAM J. Numer. Anal., 10(2), pp. 241–256. [CrossRef]
González, F. , and Kövecses, J. , 2013, “ Use of Penalty Formulations in Dynamic Simulation and Analysis of Redundantly Constrained Multibody Systems,” Multibody Syst. Dyn., 29(1), pp. 57–76. [CrossRef]
Shabana, A. A. , 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, New York.
González, F. , Dopico, D. , Pastorino, R. , and Cuadrado, J. , 2016, “ Behaviour of Augmented Lagrangian and Hamiltonian Methods for Multibody Dynamics in the Proximity of Singular Configurations,” Nonlinear Dyn., 85(3), pp. 1491–1508. [CrossRef]
Lehoucq, B. , Sorensen, D. C. , and Vu, P. , 1995, “ ARPACK: An Implementation of the Implicitly Re-Started Arnoldi Iteration That Computes Some of the Eigenvalues and Eigenvectors of a Large Sparse Matrix,” available from netlib@ornl.gov under the directory ScaLAPACK.
Flores, P. , Machado, M. , Seabra, E. , and Tavares da Silva, M. , 2011, “ A Parametric Study on the Baumgarte Stabilization Method for Forward Dynamics of Constrained Multibody Systems,” ASME J. Comput. Nonlinear Dyn., 6(1), p. 011019. [CrossRef]


Grahic Jump Location
Fig. 1

An N-loop four-bar linkage with spring elements

Grahic Jump Location
Fig. 2

A flexible double pendulum

Grahic Jump Location
Fig. 3

Coordinates used to describe the motion of each link

Grahic Jump Location
Fig. 4

A singular static equilibrium configuration for the one-loop four-bar linkage with kf = 0

Grahic Jump Location
Fig. 5

Sparsity pattern of the mass matrix of the flexible double pendulum with np = 5: (a) MCS: original Mr, (b) MCS: factorized Mr, (c) RCS: original Mq, (d) RCS: factorized Mq, (e) UCS: original Mp, and (f) UCS: factorized Mp



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