Multipoint Constraints for Modeling of Machine Tool Dynamics

The dynamic properties of machine tools are frequently calculated by means of finite-element (FE) models. Usually, in a first step, the structural components, such as machine bed, slides, columns, spindle housing, spindle, and work piece, are meshed. In a second step, these components are positioned relatively to each other and are connected by joints. Usually, the joints comprise a three-dimensional spring–damper element (SDE) and constraints that connect the SDE to adjacent structural components. Commercial FE programs do rarely offer insight into the underlying constraint equations. Rather, the constraints are realized by selecting the faces or nodes to connect and the type of constraint over a graphical user interface. Moreover, when insight into the underlying equations is offered, it is normally difficult to implement user-defined constraint equations. So far, literature lacks a coherent and in-depth description of constraints that are used for assembly of machine tool FE components. This drawback is addressed here. Different common constraints are revisited while particular focus is put on simulating moving machine axes. Common multipoint constraints (MPC) are supplemented by a shape function based node weighting. Thus, two new MPC are introduced, which improve model quality for ball screw joints (named node-to-beam (NB)-constraint) and linear guides (named RBE4-constraint). A three-axis milling machine serves as an application example for the different constraints. Simulation results are compared to experimentally derived results. Both, frequency response functions (FRF) and time-domain forced responses are considered. Showing reasonable correlation, the comparison of simulation and experiment indicates the validity of the constraints that have been introduced.