The quaternion-based three-dimensional beam theory

Computer Methods in Applied Mechanics and Engineering

Volumn 198, Issue 49-52, 2009, Pages 3944-3956

  Zupan, E ^a   Saje, M ^a   Zupan, D ^a  

^a UNIVERSITY OF LJUBLJANA   (Slovenia)

Author keywords

Interpolation of rotations; Quaternions; Three dimensional beam; Three dimensional rotation

Indexed keywords

BEAM THEORIES; CONSISTENCY CONDITIONS; FINITE ELEMENT FORMULATIONS; GOVERNING EQUATIONS; INTERNAL FORCES; MOMENT VECTOR; NUMERICAL EXAMPLE; NUMERICAL SOLUTION; PARAMETRIZATIONS; QUATERNION ALGEBRA; QUATERNIONS; THREE-DIMENSIONAL BEAM; THREE-DIMENSIONAL ROTATION; WEAK FORM;

INTERPOLATION; THREE DIMENSIONAL;

ROTATION;

EID: 70350149328     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2009.09.002     Document Type: Article

References (37)

1. Argyris J.H. An excursion into large rotations. Comput. Methods Appl. Mech. Engrg. 32 (1982) 85-155
2. Argyris J.H., Dunne P.C., Malejannakis G., and Scharpf D.W. On large displacements - small strain analysis of structures with rotational degrees of freedom. Comput. Methods Appl. Mech. Engrg. 14 (1978) 99-135
3. Argyris J., and Poterasu V.F. Large rotations - revisited application of Lie algebra. Comput. Methods Appl. Mech. Engrg. 103 (1993) 11-42
4. Atluri S.N., and Cazzani A. Rotations in computational solid mechanics. Arch. Comput. Methods Engrg. 2 (1995) 49-138
5. Bathe K.J., and Bolourchi S. Large displacement analysis of three-dimensional beam structures. Int. J. Numer. Methods Engrg. 14 (1979) 961-986
6. Battini J., and Pacoste C. Co-rotational beam elements with warping effects in instability problems. Comput. Methods Appl. Mech. Engrg. 191 (2002) 1755-1789
7. Betsch P., and Steinmann P. Frame-indifferent beam finite elements based upon the geometrically exact beam theory. Int. J. Numer. Methods Engrg. 54 (2002) 1775-1788
8. Cardona A., and Géradin M. A beam finite element non-linear theory with finite rotations. Int. J. Numer. Methods Engrg. 26 (1988) 2403-2438
9. Crivelli L.A., and Felippa C.A. The three-dimensional non-linear Timoshenko beam based on the core-congruential formulation. Int. J. Numer. Methods Engrg. 36 (1993) 3647-3673
10. Géradin M., and Rixen D. Parametrization of finite rotations in computational dynamics: a review. Revue Européenne des Éléments Finis 4 (1995) 497-553
11. Ghosh S., and Roy D. A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization. Comput. Mech. 44 (2009) 103-118
12. Ibrahimbegović A. On finite element implementation of geometrically nonlinear Reissner's beam theory: three-dimensional curved beam elements. Comput. Methods Appl. Mech. Engrg. 122 (1995) 11-26
13. Ibrahimbegović A. On the choice of finite rotation parameters. Comput. Methods Appl. Mech. Engrg. 149 (1997) 49-71
14. Iura M., and Atluri S.N. On a consistent theory and variational formulation of finitely stretched and rotated 3-D space-curved beams. Comput. Mech. 4 (1989) 73-88
15. Jelenić G., and Crisfield M.A. Geometrically exact 3D beam theory: implementation of a strain-invariant finite element for statics and dynamics. Comput. Methods Appl. Mech. Engrg. 171 (1999) 141-171
16. Jelenić G., and Saje M. A kinematically exact space finite strain beam model-finite element formulation by generalized virtual work principle. Comput. Methods Appl. Mech. Engrg. 120 (1995) 131-161
17. The MathWorks, Inc., MATLAB, Using MATLAB, Natick, 1999, .
18. McRobie F.A., and Lasenby J. Simo-Vu Quoc rods using Clifford algebra. Int. J. Numer. Methods Engrg. 45 (1999) 377-398
19. MacNeal R.H., and Harder R.L. A proposed standard set of problems to test finite element accuracy. Finite Elem. Anal. Des. 1 (1985) 3-20
20. Nour-Omid B., and Rankin C.C. Finite rotation analysis and consistent linearization using projectors. Comput. Methods Appl. Mech. Engrg. 93 (1991) 353-384
21. Omar M.A., and Shabana A.A. A two-dimensional shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243 (2001) 565-576
22. Phillips W.F., Hailey C.E., and Gebert G.A. Review of attitude representations used for aircraft kinematics. J. Aircraft 38 (2001) 718-737
23. Poreous I.R. Clifford Algebras and the Classical Groups (1995), Cambridge University Press
24. Reissner E. On finite deformation of space-curved beams. J. Appl. Math. Phys. 32 (1981) 734-744
25. Romero I. The interpolation of rotations and its application to finite element models of geometrically exact rods. Comput. Mech. 34 (2004) 121-133
26. Saje M., and Srpčič S. Large deformations of thin curved plane beam of constant initial curvature. Int. J. Mech. Sci. 28 (1986) 275-287
27. Shoemake K. Animating rotation with quaternion curves. Comput. Graph. (ACM) 19 (1985) 245-254
28. Simo J.C. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Comput. Methods Appl. Mech. Engrg. 49 (1985) 55-70
29. Simo J.C., and Vu-Quoc L. A three-dimensional finite-strain rod model. Part II: Computational aspects. Comput. Methods Appl. Mech. Engrg. 58 (1986) 79-116
30. Spurrier R.A. Comment on Singularity-free extraction of a quaternion from a direction-cosine matrix. J. Spacecraft 15 (1978) 255
31. Timoshenko S.P., and Gere J.M. Theory of Elastic Stability (1961), McGraw-Hill, New York
32. Ward J.P. Quaternions and Cayley Numbers (1997), Kluwer Academic Publishers
33. Yun X., and Bachmann E.R. Design, implementation, and experimental results of a quaternion-based Kalman filter for human body motion tracking. IEEE Trans. Robot. 22 (2006) 1216-1227
34. Zupan D., and Saje M. Finite-element formulation of geometrically exact three-dimensional beam theories based on interpolation of strain measures. Comput. Methods Appl. Mech. Engrg. 192 (2003) 5209-5248
35. Zupan D., and Saje M. On "A proposed standard set of problems to test finite element accuracy": the twisted beam. Finite Elem. Anal. Des. 40 (2004) 1445-1451
36. Zupan D., and Saje M. The linearized three-dimensional beam theory of naturally curved and twisted beams: the strain vectors formulation. Comput. Methods Appl. Mech. Engrg. 195 (2006) 4557-4578
37. E. Zupan, M. Saje, The influence of parametrization on integrating rotations from angular velocities, in: Proceedings of the Sixth International Conference on Engineering Computational Technology, September 2008, Civil-Comp Press, Stirling, 2008, .