Keywords: large rotations, co-rotational formulations, static modes, linear elastodynamics, nonlinear deformations.

The use of the finite element method in the framework of multibody dynamics allows not only the description of the large gross motion of the system components but also their deformations. This paper reviews the finite element based approaches to describe flexible multibody systems. The description of the general motion and deformation of a flexible body is obtained using an updated Lagrangean formulation [1]. As a result of the lack of the uniqueness of the displacement field a set of reference conditions is applied. The use of a fixed body, mean axis [2] and principal axis [3] reference conditions are overviewed and the benefits and drawbacks of each one of them are discussed.

In the case of nonlinear deformations it is shown that the use of absolute nodal coordinates, instead of the traditional local nodal coordinates, for the finite element description leads to a much simpler form of the equations of motion. In case of small and linear flexible body deformations the mode component synthesis is used to reduce the number of equations of motion. The selection of the modal basis to describe the flexibility of the bodies must not only describe the relevant deformations of the system components but also must be compatible with the reference conditions. In general, the authors have a preference to the use of free-free modes for the modal basis of the flexible bodies. The use of the mean axis conditions is the set of reference conditions that leads to better numerical robustness and computational efficiency.

The formulation of kinematic joints in the context of flexible multibody systems is also discussed by applying the virtual body concept [4]. It is shown that, in this form, the kinematic joints developed in the context of rigid multibody systems can still be used for flexible bodies. Moreover, by allowing the presence of deformation elements in the joints it is possible to model imperfections of such joints [5]. Several examples are introduced to demonstrate the application of the different approaches.

[1]

J. Ambrósio, "Dynamics of Structures Undergoing Gross Motion and Nonlinear Deformations: A Multibody Approach", Computers & Structures, 59(6), 1001-1012, 1996. doi:10.1016/0045-7949(95)00349-5

[2]

R.K. Cavin, A.R. Dusto, "Hamilton's Principle: Finite Element Method and Flexible Body Dynamics", AIAA Journal, 15(12), 1684-1690, 1977. doi:10.2514/3.7473

[3]

P. Nikravesh, Y.-S. Lin, "Body Reference Frames in Deformable Multibody Systems", International Journal for Multiscale Computational Engineering, 1, 1615-1683, 2003. doi:10.1615/IntJMultCompEng.v1.i23.60

[4]

J. Ambrósio, "Efficient Kinematic Joints Descriptions For Flexible Multibody Systems Experiencing Linear And Non-linear Deformations", Int. J. of Numerical Methods in Engineering, 56, 1771-1793, 2003. doi:10.1002/nme.639

[5]

P. Verissimo, J. Ambrósio, "Improved bushing models for general multibody systems and vehicle dynamics", Multibody System Dynamics, 22(4), 341-365, 2009. doi:10.1007/s11044-009-9161-7

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