Modelling of Moving Interfaces for Reduced-Order Finite Element Models using Trigonometric Interpolation

Abstract

Flexible multi-body simulation by means of reduced-order finite element models is gathering in importance for the simulation of mechanical and mechatronic systems like, e.g.\ machine tools or handling systems. Moving of the system's axes involves changing of the coupling position between flexible bodies, and thus, changing of the finite element nodes involved in an interface. Because modern model reduction techniques are based on matching properties of the system's input-output behaviour, the order of the reduced model strongly depends on the number of interfaces used. Therefore, it is not appropriate to use all finite element nodes of a potential interface area as independent inputs. The subject of this paper is a method for modelling of moving interfaces to flexible bodies which is compatible with model order reduction. First, a formalism is presented which allows the application of distributed loads onto the nodal degrees of freedom of finite element nodes. Next, elementary load distributions which allow the application of forces and torques to a surface are introduced and, in a further step, orthonormalised in order to achieve stationary interfaces which provide the desired six degrees of freedom with an arbitrary center of action. Subsequently, a method using trigonometric interpolation of a weighting function is presented which enables modelling of interfaces moving along a predefined path and acting on a known set of surfaces. As weighting function for geometrical restriction of the action on a surface, a trapezoidal function is suggested. As with the stationary interfaces, elementary load distributions for moving interfaces are presented and orthonormalised, what allows modelling of moving interfaces with six degrees of freedom. The resulting nodal degrees of freedom are visualised by means of examples and analysed for different meshes. For appropriate finite element meshes, the maximum relative error of action lies below $10^{-5}$ and the maximum cross-coupling between the interface's degrees of freedom lies below $10^{-3}$ for the majority of the moving path length. Due to the trigonometric interpolation approach, only a low number of harmonics are to be used as interface matrices for the finite element system what qualifies the method for the use in combination with model order reduction.