Computation of distributed forces in modally reduced mechanical systems

The consideration of state depended and distributed loads, like contact and friction forces, can be of remarkable computational effort in the framework of a modally reduced mechanical system. This is caused by the common strategy, in which such forces are expressed in the coordinates of the high dimensional unreduced space. In case of a Finite Element (FE) model the latter approach involves three steps. First, the physical degrees of freedom (DOF) are determined out of the DOF of the reduced (modal) system. In a subsequent step, the FE node forces have to be computed based on certain physical laws in the high dimensional space of the FE model. Finally, the reduced force vector is determined by the projection of the physical force vector into the modal space. This paper is devoted to a more efficient computation of such loads. It will be shown that a consequent computation in the reduced (modal) space requires the well known displacement trial vectors (commonly called 'modes') as well as 'force trial vectors', which we will denote as 'force - modes'. It will be shown that any base of displacement trial vectors leads to an associated base of force trial vectors and all system relevant loads can be computed by a superposition of these force-modes. Consequently, the force computation can be done in the low dimensional space of the reduced system. This paper is subdivided into three main sections. In the first part, the motivation for this research project is outlined. In the second section the theory is presented and finally a numerical example from elastohydrodynamics demonstrates the methods potential.