Non-linear dynamics of flexible multibody systems

Abstract In this work we set to examine several important issues pertinent to currently very active research area of the finite element modeling of flexible multibody system dynamics. To that end, we first briefly introduce three different model problems in non-linear dynamics of flexible 3D solid, a rigid body and 3D geometrically exact beam, which covers the vast majority of representative models for the particular components of a multibody system. The finite element semi-discretization for these models is presented along with the time-discretization performed by the mid-point scheme. In extending the proposed methodology to modeling of flexible multibody systems, we also present how to build a systematic representation of any kind of joint connecting two multibody components, a typical case of holonomic contraint, as a linear superposition of elementary constraints. We also indicate by a chosen model of rolling contact, an example of non-holonomic constraint, that the latter can also be included within the proposed framework. An important aspect regarding the reduction of computational cost while retaining the consistency of the model is also addressed in terms of systematic use of the rigid component hypothesis, mass lumping and the appropriate application of the explicit–implicit time-integration scheme to the problem on hand. Several numerical simulations dealing with non-linear dynamics of flexible multibody systems undergoing large overall motion are presented to further illustrate the potential of presented methodology. Closing remarks are given to summarize the recent achievements and point out several directions for future research.

[1]  J. W. Humberston.  Classical mechanics , 1980, Nature.

[2]  E. Reissner,et al.  On One‐Dimensional Large‐Displacement Finite‐Strain Beam Theory , 1973 .

[3]  Alberto Cardona,et al.  Rigid and flexible joint modelling in multibody dynamics using finite elements , 1991 .

[4]  Boštjan Brank,et al.  Stress resultant geometrically exact form of classical shell model and vector‐like parameterization of constrained finite rotations , 2001 .

[5]  J. C. Simo,et al.  Non-linear dynamics of three-dimensional rods: Exact energy and momentum conserving algorithms , 1995 .

[6]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[7]  Adnan Ibrahimbegovic,et al.  Energy conserving/decaying implicit time-stepping scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations , 2002 .

[8]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .

[9]  T. R. Kane,et al.  Dynamics of a cantilever beam attached to a moving base , 1987 .

[10]  A. Ibrahimbegovic.  On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements , 1995 .

[11]  Mark W. Spong,et al.  Robot dynamics and control , 1989 .

[12]  B. J. Hsieh,et al.  Non-Linear Transient Finite Element Analysis with Convected Co--ordinates , 1973 .

[13]  A. Ibrahimbegovic,et al.  Finite rotations in dynamics of beams and implicit time-stepping schemes , 1998 .

[14]  K. Tamma,et al.  A robust self‐starting explicit computational methodology for structural dynamic applications: Architecture and representations , 1990 .

[15]  K. Bathe,et al.  Large displacement analysis of three‐dimensional beam structures , 1979 .

[16]  A. J. Chen,et al.  Finite Element Method in Dynamics of Flexible Multibody Systems: Modeling of Holonomic Constraints and Energy Conserving Integration Schemes , 2000 .

[17]  J. C. Simo,et al.  Unconditionally stable algorithms for rigid body dynamics that exactly preserve energy and momentum , 1991 .

[18]  A. Ibrahimbegovic.  On the choice of finite rotation parameters , 1997 .

[19]  A. Ibrahimbegovic.  Stress Resultant Geometrically Exact Shell Theory for Finite Rotations and Its Finite Element Implementation , 1997 .

[20]  Ronald L. Huston,et al.  Multibody Dynamics — Modeling and Analysis Methods , 1991 .

[21]  J. Huetink,et al.  Progress in mixed Eulerian-Lagrangian finite element simulation of forming processes , 1990 .

[22]  Michał Kleiber,et al.  Computational aspects of nonlinear structural systems with large rigid body motion , 2001 .

[23]  R. Taylor,et al.  An explicit-implicit method for flexible-rigid multibody systems , 2001 .

[24]  Ted Belytschko,et al.  Mixed methods for time integration , 1979 .

[25]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[26]  Kumar K. Tamma,et al.  An explicit velocity based Lax-Wendroff/Taylor-Galerkin methodology of computation for the dynamics of structures , 1988 .

[27]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[28]  B. D. Veubeke,et al.  The dynamics of flexible bodies , 1976 .

[29]  J. C. Simo,et al.  The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics , 1992 .

[30]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[31]  F. Armero,et al.  On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics , 2001 .

[32]  J. Z. Zhu,et al.  The finite element method , 1977 .

[33]  Charis J. Gantes,et al.  Deployable Structures : Analysis and Design , 2001 .

[34]  J. Argyris.  An excursion into large rotations , 1982 .

[35]  A. Ibrahimbegovic,et al.  On rigid components and joint constraints in nonlinear dynamics of flexible multibody systems employing 3D geometrically exact beam model , 2000 .

[36]  Thomas J. R. Hughes,et al.  Implicit-explicit finite elements in nonlinear transient analysis , 1979 .