Abstract
This paper presents general variational formulations for dynamical problems, which are easily implemented numerically. The development presents the relationship between the very general weak formulation arising from linear and angular momentum balance considerations, and well known variational priciples. Two and three field mixed forms are developed from the general weak form. The variational principles governing large rotational motions are linearized and implemented in a time finite element framework, with appropriate expressions for the relevant “tangent” operators being derived. In order to demonstrate the validity of the various formulations, the special case of free rigid body motion is considered. The primal formulation is shown to have unstable numerical behavior, while the mixed formulation exhibits physically stable behavior. The formulations presented in this paper form the basis for continuing investigations into constrained dynamical systems and multi-rigid-body systems, which will be reported in subsequent papers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Agrawal O. P.; ShabanaA. A. (1986): Automated visco-elastic analysis of large scale inertia-variant spacial vehicles. Comput. Struct. 22, 165–178
Atluri S. N.; Amos A. K. (1987): Large space structures: dynamics and control. Berlin, Heidelberg, New York: Springer (Springer series comp. mechanics)
Bailey C. D. (1975): Application of hamilton's law of varying action. AIAA J. 13, 1154–1157
Banerjee A. K. (1987): Comment on “relationship between Kane's equations and the Gibbs-Appell equations”. J. Guid. Control Dyn. 10, 596
Baruch M.; Riff R. (1982): Hamilton's principle, Hamilton's law-6n correct formulations. AIAA J. 20, 687–692
Belytschko T.; Hughes T. J. R. (1981). Computational methods in transient analysis. Amsterdam: North-Holland Publishing
Borri M.; Ghiringhelli G. L.; Lans M.; Mantegarra P.; Merlini T. (1985): Dynamic response of mechanical systems by a weak Hamiltonian formulation. Comput. Struct. 20, 495–508
Desloge E. A. (1987): Relationship between Kane's equations and the Gibbs-Appell equations. J. Guid. Control Dyn. 10, 120–122
Geradin M.; Cardona A. (1989): Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra. Comput. Mech. 4, 115–135
Haug E. J.; Wu S. C.; Yang S. M. (1986): Dynamics of mechanical systems with coulomb friction, stiction, impact and constraint addition-deletion — I theory. Mech. Mach. Theory 21, 401–406
Haug E. J.; McCullough M. K. (1986): A variational — vector calculus approach to machine dynamics. J. Mech. Trans. Automat. Design 108, 25–30
Hughes P. C. (1986). Spacecraft attitude dynamics. New York: Wiley
Iura M.; AtluriS. N. (1989): On a consistent theory and variational formulation of finitely stretched and rotated 3-d space-curved beams. Comput. Mech. 4, 73–88
Kane T. R.; LevinsonD. A. (1980): Formulation of equations of motion for complex space-craft. J. Guid. Control Dyn. 3, 99–112
Kane T. R.; Likins P. W.; Levinson D. A. (1983). Spacecraft dynamics. New York: McGraw-Hill
Kardestuncer H. (1987). Finite element handbook. New York: McGraw-Hill
Khulief Y. A.; Shabana A. A. (1986): Dynamics of multibody systems with variable kinematic structure. J. Mech. Trans. Automat. Design 108, 167–175
Kim S. S.; Shabana A. A.; Haug E. J. (1984): Automated Vehicle Dynamic Analysis with Flexible Components. J. Mech. Trans. Automat. Design 106, 126–132
Malkus D. S.; Hughes T. J. R. (1978): Mixed finite element methods-reduced and selective integration techniques: A unification of concepts. Comput. Methods Appl. Mech. Eng. 15, 63–81
McCullough M. K.; Haug E. J. (1986): Dynamics of high mobility track vehicles. J. Mech. Trans. Automat. Design 108, 189–196
Meirovitch L.: Quinn R. D. (1987): Equations of motion for maneuvering flexible space-craft. J. Guid. Control Dyn. 10, 453–465
Modi V. J.; Ibrahim A. M. (1987). On the transient dynamic analysis of flexible orbiting structures. Large space structures: dynamics and control. Berlin, Heidelberg, New York: Springer
Pietraszkiewicz W.; Badur J. (1983): Finite rotations in the description of continuum deformation. Int. J. Eng. Sci. 21, 1097–1115
Shi, G. (1988): Nonlinear static and dynamic analyses of large-scale lattice-type structures and nonlinear active control by Piezo actuators. Ph.D. Thesis, Georgia Institute of Technology
Smith D. R.; Smith C. V. (1974): When is Hamilton's principle an extremum principle? AIAA J. 12, 1573–1576
Struelpnagel J. (1964): On the parametrization of the three-dimensional rotation group. SIAM Rev. 6, 422–430
Washizu K. (1980). Variational methods in elasticity and plasticity. 3rd ed. New York: Pergamon Press
Wittenburg, J.; Wolz, U. (1985): MESA VERDE: A symbolic program for nonlinear articulated rigid body dynamics. ASME 85-DET-151
Zienkiewicz O. C.; Taylor R. L.; Too J. M. (1971): Reduced integration technique in general analysis of plates and shells. Int. J. Num. Methods Eng. 3, 275–290
Author information
Authors and Affiliations
Additional information
Communicated by G. Yagawa, October 1, 1989
Rights and permissions
About this article
Cite this article
Borri, M., Mello, F. & Atluri, S.N. Variational approaches for dynamics and time-finite-elements: numerical studies. Computational Mechanics 7, 49–76 (1990). https://doi.org/10.1007/BF00370057
Issue Date:
DOI: https://doi.org/10.1007/BF00370057