Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory

The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.

[1]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[2]  H. Ashley.  Observations on the dynamic behavior of large flexible bodies in orbit. , 1967 .

[3]  Irving H. Shames,et al.  Solid mechanics: a variational approach , 1973 .

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

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

[6]  P. Likins,et al.  Floating reference frames for flexible spacecraft , 1977 .

[7]  D. W. Scharpf,et al.  Finite element method — the natural approach , 1979 .

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

[9]  W. L. Burke.  Applied Differential Geometry , 1985 .

[10]  C. Rankin,et al.  An element independent corotational procedure for the treatment of large rotations , 1986 .

[11]  J. C. Simo,et al.  On the Dynamics of Flexible Beams Under Large Overall Motions—The Plane Case: Part II , 1986 .

[12]  M. Greenberg.  Advanced Engineering Mathematics , 1988 .

[13]  O. Bauchau,et al.  A Multibody Formulation for Helicopter Structural Dynamic Analysis , 1993 .

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

[15]  Ahmed A. Shabana,et al.  APPLICATION OF THE ABSOLUTE NODAL CO-ORDINATE FORMULATION TO MULTIBODY SYSTEM DYNAMICS , 1997 .

[16]  A. Shabana.  Computer Implementation of the Absolute Nodal Coordinate Formulation for Flexible Multibody Dynamics , 1998 .

[17]  Ahmed A. Shabana,et al.  Application of the Absolute Nodal Coordinate Formulation to Large Rotation and Large Deformation Problems , 1998 .

[18]  Ahmed A. Shabana,et al.  Use of Cholesky Coordinates and the Absolute Nodal Coordinate Formulation in the Computer Simulation of Flexible Multibody Systems , 1999 .