A new floating frame of reference formulation for flexible multibody dynamics

For the dynamic analysis of flexible multibody systems, three conceptually different descriptions are available and commonly used: the inertial frame formulations, the corotational formulations and the floating frame formulations. The differences between these formulations are mainly in the way they describe a body’s elastic behavior or, in particular, in the degrees of freedom used to describe this elastic behavior. As a consequence, differences occur in the way kinematic constraints between bodies are enforced. The inertial frame and corotational frame formulations can both be interpreted as nonlinear finite element methods. As such, they have in common that they use the absolute nodal coordinates as degrees of freedom. Constraints between bodies can be enforced by simply equating the relevant degrees of freedom that both bodies share at an interface point. On the other hand, the floating frame formulations use the absolute floating frame of reference coordinates together with a set of local generalized coordinates that describe a body’s local elastic displacement field as degrees of freedom. Because the absolute interface coordinates are not part of the degrees of freedom, constraints are enforced using Lagrange multipliers. This increases the total number of unknowns and causes the constrained equations of motion to be of the differential-algebraic type. In this presentation, an overview is given of a newly developed floating frame of reference formulation of which the details are explained in [1]. In this new method, the absolute interface coordinates are used as degrees of freedom. To this end, coordinate transformations are developed that express the absolute floating frame coordinates and the local generalized coordinates in terms of the absolute interface coordinates. Not only does the new method not require the use of Lagrange multipliers for enforcing constraints, it also offers the possibility to reduce geometric nonlinear systems by applying important model order reduction techniques in a body’s local frame. Using the well-developed Craig-Bampton method, it is possible to create so-called superelements in a flexible multibody formulation. That is, for each flexible body, the tangent mass and stiffness matrices, reduced to the interface points, can be obtained from linear finite element models and can directly be applied in the dynamic analysis of the entire system. Reference [1] M.H.M. Ellenbroek, J.P. Schilder: On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics. Multibody System Dynamics, submitted 03-05-2017

Flexible multibody dynamics: Superelements using absolute interface coordinates in the floating frame formulation

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

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems

Efficient modelling of short and wide leaf springs using a superelement formulation

In this work, a recently developed superelement in the floating frame of reference formulation will be applied to plate elements. It is shown that for applications in which short and wide leaf springs are required, currently used flexible multibody models using beam elements do not provide sufficient accuracy. Relevance of the new formulation for the efficient modelling of flexure mechanisms will be shown