flexible multibody dynamics - superelements using absolute interface coordinates in the floating frame formulation
Jurnan Schilder is a PhD student in the department of Mechanics of Solids, Surfaces & Systems (MS3). His supervisor is prof.dr.ir. A. de Boer from the faculty of Engineering Technology (ET).
The floating frame formulation is a well-established and widely used formulation in flexible multibody dynamics. In this formulation the rigid body motion of a flexible body is described by the absolute generalized coordinates of the body’s floating frame with respect to the inertial frame. The body’s flexible behavior is described locally, relative to the floating frame, by a set of deformation shapes. Because in many situations, the elastic deformations of a body remain small, these deformation shapes can be determined by applying powerful model order reduction techniques to a body’s linear finite element model. This is an important advantage of the floating frame formulation in comparison with for instance nonlinear finite element formulations.
An important disadvantage of the floating frame formulation is that it requires Lagrange multipliers to satisfy the kinematic constraint equations. The constraint equations are typically formulated in terms of the generalized coordinates corresponding to the body’s interface points, where it is connected to other bodies or the fixed world. As the interface coordinates are not part of the degrees of freedom of the formulation, the constraint equations are in general nonlinear equations in terms of the generalized coordinates, which cannot be solved analytically.
In this work, a new formulation is presented with which it is possible to eliminate the Lagrange multipliers from the constrained equations of motion, while still allowing the use of linear model order reduction techniques in the floating frame. This is done by reformulating a flexible body’s kinematics in terms of its absolute interface coordinates. One could say that the new formulation creates a superelement for each flexible body. These superelements are created by establishing a coordinate transformation from the absolute floating frame coordinates and local interface coordinates to the absolute interface coordinates. In order to establish such a coordinate transformation, existing formulations commonly require the floating frame to be in an interface point. The new formulation does not require such strict demands and only requires that there is zero elastic deformation at the location of the floating frame. In this way, the new formulation offers a more general and elegant solution to the traditional problem of creating superelements in the floating frame formulation.
The fact that the required coordinate transformation involves the interface coordinates, makes it natural to use the Craig-Bampton method for describing a body’s local elastic deformation. After all, the local interface coordinates equal the generalized coordinates corresponding to the static Craig-Bampton modes. However, in this work it is shown that the new formulation can deal with any choice for the local deformation shapes. Also, it is shown how the method can be expanded to include geometrical nonlinearities within a body.
A full and complete mathematical derivation of the new formulation is presented. However, an extensive effort is made to give geometric interpretation to the transformation matrices involved. In this way the new method can be understood better from an intuitive engineering perspective. This perspective has led to the proposal of several additional approximations to simplify the formulation. Validation simulations of benchmark problems have shown the new formulation to be accurate and the proposed additional approximations to be appropriate indeed.
