Abstract
The Absolute-Coordinate-Based (ACB) method that combines the Natural Coordinate Formulation (NCF) describing rigid bodies and the Absolute Nodal Coordinate Formulation (ANCF) describing flexible bodies has been widely used to study the dynamics of rigid-flexible multibody system since it exhibits many good features, such as the constancy of the mass matrix of the derived dynamic equation, and the easy description and great simplification of the constraint conditions. In order to achieve these good features, both NCF and ANCF take the vectors, rather than rotational coordinates, to describe the rotation and deformation of the rigid-flexible bodies. In this study, the physical meaning of the components of the generalized force vector corresponding to the vector coordinates is revealed on the basis of both ANCF and NCF. Some new and simple formulations of imposing moments and evaluating joint reaction forces for rigid-flexible multibody system are proposed by making full use of the physical meaning of vector coordinates and Lagrange multipliers. All the proposed formulations are defined in the global frame so as to avoid the coordinate transformation. Hence, it can be directly applicable to various types of finite elements of ANCF, including the slope deficient elements. Finally, several typical and practical examples are used to verify the effectiveness of the proposed formulations.
Similar content being viewed by others
References
Shabana, A.A.: An absolute nodal coordinates formulation for the large rotation and deformation analysis of flexible bodies. Technical Report. No. MBS96-1-UIC, University of Illinois at Chicago (1996)
Shabana, A.A.: Definition of the slopes and absolute nodal coordinate formulation. Multibody Syst. Dyn. 1, 339–348 (1997)
Eberhard, P., Schiehlen, W.: Computational dynamics of multibody systems history, formalisms, and applications. J. Comput. Nonlinear Dyn. 1, 3–12 (2006)
Yoo, W.S., Dmitrochenko, O., Yu, D.: Review of finite elements using absolute nodal coordinates for large-deformation problems and matching physical experiments. In: ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DETC2005-84720, Long Beach, CA (2005)
Schiehlen, W.: Research trends in multibody system dynamics. Multibody Syst. Dyn. 18, 3–13 (2007)
García-Vallejo, D., Mikkola, A.M., Escalona, J.S.: A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn. 50, 249–264 (2007)
Gerstmayr, J., Matikainen, M.K., Mikkola, A.M.: A geometrically exact beam element based on the absolute nodal coordinate formulation. Multibody Syst. Dyn. 20, 359–384 (2008)
Tian, Q., Zhang, Y., Chen, L., Flores, P.: Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Comput. Struct. 87, 913–929 (2009)
Tian, Q., Zhang, Y., Chen, L., Yang, J.: Simulation of planar flexible multibody systems with clearance and lubricated revolute joints. Nonlinear Dyn. 60, 489–511 (2010)
Abdel-Nasser, A.M., Shabana, A.A.: A nonlinear visco-elastic constitutive model for large rotation finite element formulations. Multibody Syst. Dyn. 26, 57–79 (2011)
García De Jalón, J., Bayo, E.: Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge. Springer, New York (1994)
García De Jalón, J.: Twenty-five years of natural coordinates. Multibody Syst. Dyn. 18, 15–33 (2007)
García-Vallejo, D., Escalona, J.L., Mayo, J., Domínguez, J.: Describing rigid-flexible multibody systems using absolute coordinates. Nonlinear Dyn. 34, 75–94 (2003)
García-Vallejo, D., Mayo, J., Escalona, J.L., Domínguez, J.: Three-dimensional formulation of rigid-flexible multibody systems with flexible beam elements. Multibody Syst. Dyn. 20, 1–28 (2008)
Tian, Q., Liu, C., Machado, M., Flores, P.: A new model for dry and lubricated cylindrical joints with clearance in spatial flexible multibody systems. Nonlinear Dyn. 64, 25–67 (2011)
Liu, C., Tian, Q., Hu, H.Y.: Dynamics of a large scale rigid-flexible multibody system composed of composite laminated plates. Multibody Syst. Dyn. 26, 283–305 (2011)
García De Jalón, J., Callejo, A.: A straight methodology to include multibody dynamics in graduate and undergraduate subjects. Mech. Mach. Theory 46, 168–182 (2011)
Betsch, P., Stein, E.: An assumed strain approach avoiding artificial thickness straining for a nonlinear 4-node shell element. Commun. Numer. Methods Eng. 11, 899–909 (1995)
Omar, M.A., Shabana, A.A.: A two-dimension shear deformable beam for large rotation and deformation problems. J. Sound Vib. 243, 565–576 (2001)
Sopanen, J.T., Mikkola, A.M.: Description of elastic forces in absolute nodal coordinate formulation. Nonlinear Dyn. 34, 53–74 (2003)
Gerstmayr, J., Shabana, A.A.: Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dyn. 45, 109–130 (2006)
Dufva, K., Shabana, A.A.: Analysis of thin plate structures using the absolute nodal coordinate formulation. Proc. Inst. Mech. Eng., Proc., Part K, J. Multi-Body Dyn. 219, 345–355 (2005)
Romero, I., Arribas, J.J.: A simple method to impose rotations and concentrated moments on ANC beams. Multibody Syst. Dyn. 21, 307–323 (2009)
Shabana, A.A., Yakoub, R.Y.: Three-dimensional absolute nodal coordinate formulation for beam elements: theory. J. Mech. Des. 123, 606–613 (2001)
Yakoub, R.Y., Shabana, A.A.: Three dimensional absolute nodal coordinate formulation for beam elements: implementation and applications. J. Mech. Des. 123, 614–621 (2001)
Shabana, A.A.: Dynamics of Multibody Systems. Cambridge University Press, New York (2005)
Arnold, M., Brüls, O.: Convergence of the generalized-a scheme for constrained mechanical systems. Multibody Syst. Dyn. 18, 185–202 (2007)
Dufva, K.E., Sopanen, J.T., Mikkola, A.M.: Three dimensional beam element based on a cross sectional coordinate system approach. Nonlinear Dyn. 43, 311–327 (2006)
Timoshenko, S., Gere, J.: Mechanics of Materials. Van Nostrand Reinhold, New York (1972)
Xu, W., Liu, Y., Liang, B., Wang, X., Xu, Y.: Unified multi-domain modeling and simulation of space robot for capturing a moving target. Multibody Syst. Dyn. 23, 293–331 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, C., Tian, Q., Hu, H. et al. Simple formulations of imposing moments and evaluating joint reaction forces for rigid-flexible multibody systems. Nonlinear Dyn 69, 127–147 (2012). https://doi.org/10.1007/s11071-011-0251-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-011-0251-8