Abstract
The reduction in the number of coordinates for flexible multibody systems is necessary in order to achieve acceptable simulation times of real-life structures and machines. The conventional model order reduction technique for flexible multibody systems is based on the floating frame of reference formulation (FFRF), using a rigid body frame and superimposed small flexible deformations. The FFRF leads to strongly coupled terms in rigid body and flexible coordinates as well as to a non-constant mass matrix. As an alternative to the FFRF, a formulation based on absolute coordinates has been proposed which uses a co-rotational strain. In this way, a constant mass matrix and a co-rotational stiffness matrix are obtained. In order to perform a reduction in the number of coordinates, by means of the component mode synthesis, e.g., the number of modes needs to be increased, such that all modes are represented in every possible rotated configuration. This approach leads to the method of generalized component mode synthesis (GCMS). The present paper shows in detail how the equations of motion of the FFRF evolve from the ones of the GCMS by considering rigid body constraint conditions and subsequently eliminating them via an appropriate null-space projection. This approach allows a straightforward, term-by-term interpretation of the FFRF mass matrix and of the generalized gyroscopic forces, which, to the same extent, cannot be deduced from former publications on the FFRF. From a practical point of view, the resulting expressions allow to calculate all inertia coefficients from the constant finite element mass matrix together with standard input data of the finite element model in the course of a preprocessing step. Then, the repeated updates of the FFRF mass matrix and of the gyroscopic forces in the course of time integration involve only simple vector matrix operations of low dimensions. In contrast to previous implementations of the FFRF, no evaluations of extra inertia integrals are required. Consequently, the present formulation can be implemented entirely independent of the related finite element code.
Highlights
Introduction and state of the artMachines, cars, planes, and other technical systems experience a continuous growth in performance
The so-called component mode synthesis (CMS) method has been developed, which is based on a small number of static and dynamic mode shapes that accurately describe the deformation of the bodies
The paper shows the interrelation of the absolute coordinates formulation (ACF) and the floating frame of reference formulation (FFRF)
Summary
The absolute coordinate formulation is based on a Lagrangian finite element (FE) formulation as known from textbooks on finite elements [17]. The spatial discretization of the displacement field is built upon the matrix of space-wise shape functions NFE(x) and the vector of nodal displacements qFE(t). With x being the spatial coordinate in the reference configuration, u the displacement field, and t the time. A full derivation of the equations of motion can be found in [10,16]. P1 eZ r1 r finite element coordinates ηj P ξj finite element mesh eY global frame P2. The vector of external forces is denoted as f ext. It is noted that the equations of motion are written for a single flexible body, while in the implementation, all bodies share a large vector of coordinates. The constraints, which are applied either to the body itself (e.g. ground constraint) or which act between bodies, need to be written as a function of the coordinates of all bodies being involved in the constraints
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