A high-precision co-rotational formulation of 3D beam elements for dynamic analysis of flexible multibody systems

Abstract Evaluating inertia forces is a central and complicated task for dynamic analysis of flexible multibody systems (FMS) involving large displacements and rotations. In a number of co-rotational approaches, cubic interpolations are adopted to formulate both inertia and internal forces, where the inertia term may be complicated in derivation and several Gauss points are needed in the numerical integration. The paper presents a high-precision co-rotational formulation of 3D beam elements for dynamic analysis of FMS. The method of switching the rotation vector and its complement is adopted to avoid the singularity problem in spatial finite rotations. In contrast to the traditional co-rotational formulation, the governing equations in the paper are formulated based on the principle of virtual power, without requiring the variation of rotation matrix. In the framework of the proposed co-rotational formulation, the stiffness matrix of small-deformation beam element can be used directly, including the Euler–Bernoulli and Timoshenko–Reissner beam elements. More essentially, the inertia terms of the beam element are formulated by discretizing the beam element into three lumped masses at the left and right end nodes, and an auxiliary node at the middle point of the beam element, resulting in a lumped mass matrix. The equivalence ensures that the total mass is completely accurate and the moment of inertia is high-precision if the beam element undergoes small elastic deformation in the local coordinate system. In the local coordinate system of co-rotational formulations, small elastic deformation can be guaranteed so that the inertia forces are formulated analytically without needing the Gauss integration. Finally, four numerical examples are considered to evaluate the accuracy of the formulation against to the geometrically exact beam theory and the previous co-rotational beam formulations. The proposed method can give high-precision numerical results from the comparison, even if using a small number of elements.

[1]  A. Shabana.  Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation , 1997 .

[2]  J. Stuelpnagel.  On the Parametrization of the Three-Dimensional Rotation Group , 1964 .

[3]  A. Shabana.  Definition of ANCF Finite Elements , 2015 .

[4]  Z. Li,et al.  A co-rotational formulation for 3D beam element using vectorial rotational variables , 2006 .

[5]  J. C. Simo,et al.  On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach , 1988 .

[6]  Jan B. Jonker,et al.  A geometrically non-linear formulation of a three-dimensional beam element for solving large deflection multibody system problems , 2013 .

[7]  Guo-Ping Cai,et al.  Dynamics studies of a flexible hub-beam system with significant damping effect , 2008 .

[8]  Hui Yang,et al.  Dynamics modelling of a flexible hub-beam system with a tip mass , 2003 .

[9]  C. Rankin,et al.  Finite rotation analysis and consistent linearization using projectors , 1991 .

[10]  M. Crisfield,et al.  Dynamics of 3-D co-rotational beams , 1997 .

[11]  J. C. Simo,et al.  A three-dimensional finite-strain rod model. Part II: Computational aspects , 1986 .

[12]  A. Mikkola,et al.  Review on the Absolute Nodal Coordinate Formulation for Large Deformation Analysis of Multibody Systems , 2013 .

[13]  A. Ibrahimbegovic,et al.  Computational aspects of vector-like parametrization of three-dimensional finite rotations , 1995 .

[14]  Ahmed A. Shabana,et al.  Use of independent rotation field in the large displacement analysis of beams , 2014 .

[15]  Lei Yu,et al.  Integration of absolute nodal elements into multibody system , 2010 .

[16]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

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

[18]  S. Atluri,et al.  Dynamic analysis of planar flexible beams with finite rotations by using inertial and rotating frames , 1995 .

[19]  Qiang Xue,et al.  Dynamic response and instability of frame structures , 2001 .

[20]  K. Hsiao,et al.  A consistent co-rotational finite element formulation for geometrically nonlinear dynamic analysis of 3-D beams , 1999 .

[21]  Qiang Xue,et al.  A study on the instability problem for 3D frames , 1998 .

[22]  P. Betsch,et al.  Constrained dynamics of geometrically exact beams , 2003 .

[23]  Tamer M. Wasfy,et al.  Computational strategies for flexible multibody systems , 2003 .

[24]  M. Hjiaj,et al.  A consistent 3D corotational beam element for nonlinear dynamic analysis of flexible structures , 2014 .

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

[26]  Jean-Marc Battini,et al.  A modified corotational framework for triangular shell elements , 2007 .

[27]  M. A. Crisfield,et al.  A unified co-rotational framework for solids, shells and beams , 1996 .

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

[29]  M. Hjiaj,et al.  Dynamics of 3D beam elements in a corotational context: A comparative study of established and new formulations , 2012 .

[30]  M. Géradin,et al.  A beam finite element non‐linear theory with finite rotations , 1988 .

[31]  Arend L. Schwab,et al.  Dynamics of Multibody Systems , 2007 .

[32]  Jiabei Shi,et al.  A New Rotation-Free Shell Formulation Using Exact Corotational Frame for Dynamic Analysis and Applications , 2018 .

[33]  Carlos A. Felippa,et al.  A unified formulation of small-strain corotational finite elements: I. Theory , 2005 .

[34]  Gang Wang,et al.  Geometrical nonlinear and stability analysis for slender frame structures of crawler cranes , 2015 .

[35]  M. Crisfield.  A consistent co-rotational formulation for non-linear, three-dimensional, beam-elements , 1990 .

[36]  Edouard Rivière-Lorphèvre,et al.  Modelling of flexible bodies with minimal coordinates by means of the corotational formulation , 2018 .

[37]  J. Mäkinen.  Total Lagrangian Reissner's geometrically exact beam element without singularities , 2007 .

[38]  M. Hjiaj,et al.  Efficient formulation for dynamics of corotational 2D beams , 2011 .

[39]  Fumio Fujii,et al.  An explicit algorithm for geometrically nonlinear transient analysis of spatial beams using a corotational total Lagrangian finite element formulation , 2018 .

[40]  Jiabei Shi,et al.  Dynamic contact model of shell for multibody system applications , 2018, Multibody System Dynamics.

[41]  Debasish Roy,et al.  A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization , 2009 .

[42]  Ahmed A. Shabana,et al.  Flexible Multibody Dynamics: Review of Past and Recent Developments , 1997 .

[43]  Pinqi Xia,et al.  Rotation vector and its complement parameterization for singularity-free corotational shell element formulations , 2019, Computational Mechanics.

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

[45]  Qiang Tian,et al.  Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods , 2009 .

[46]  C. Pacoste,et al.  Co-rotational beam elements with warping effects in instability problems , 2002 .

[47]  Gordan Jelenić,et al.  Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics , 1999 .