5 Workpiece Coupling in Machine Tools Using Experimental-Analytical Dynamic Substructuring 51 Fig. 5.2 Modelling of workpiece and machine table (left) and reference system and its vibration modes in z-direction (right) Aim of the analysis is to implement the proposed method for the simulated structures for predicting the target FRF of 5z5z in the assembled state. For this, firstly, an FE model of the unconstrained (free-free) and reduced workpiece is created. It has four points of attachment (1–4) and one internal point (5). An interface condensation node represents each point of attachment. The condensation node is connected to multiple surface nodes within a circular section with rigid body elements (RBE3 in Siemens NX). The radius of the circular section is calculated based on the radius of the pressure cone of the bolted joint [1]. Subsequently, free-free FRFs are calculated for every combination of excitation and response DOFs at the five nodes which include six DOFs each at the interface nodes and one DOF at the internal node (5z). Thus, a full scale response model of the workpiece is obtained. Similarly, the response model of the discretized table at the four interface points is also achieved. For analyzing the effect of including and ignoring particular DOFs in the coupling calculations, four cases were created. Since the target FRF is in the translational z-direction, it is obvious that compliance in this direction at interface nodes should be considered in the coupling calculations for all cases. In the first case, only the translational DOFs in z at each interface node of the workpiece and table were chosen for coupling. This includes also the cross compliances between the nodes and results is a (BGBT)−1 matrix of size 4 ×4 in Eq. (5.10). From cases 2 to 4, the translational DOF z as well as a rotational compliance about each axis were selected (z and θx, z and θy, z and θz). The matrix (BGBT)−1 now has a size of 8 ×8 for each coupling case. The amplitude and phase plots of the target FRF (5z5z) for the coupling cases can be observed in Fig. 5.3. As can be seen for most of the cases, an exact match of the reference system could not be achieved. This is mainly because only one or two coupling DOFs are considered and all other DOFs and their cross compliance are ignored. Additionally, the difference in the mesh size of the master nodes of the workpiece and table meshes and the model order reduction could contribute to some of the deviation. In case 1, the first resonance peak at 44 Hz originating form the tilting of the table bout x-axis could be estimated well. A significant error in the predicted frequency of the second resonance peak at 2250 Hz is observable. This is also be seen in cases 2 and 4. The reason is thought to be that the bending mode of the workpiece has a smaller component in the z, θx and θz. On the other hand, a very good estimation is achieved when z and θy are selected as coupling DOFs. The first resonance pole shows some deviation but the zero at ca. 586 Hz and especially the pole at 2250 Hz could be estimated well. This can be attributed to the very good observability of the workpiece bending mode in the rotational compliance about the y-axis, θy. Thus, it can be observed that the consideration of merely one translational coupling DOF, as done in previous literature [12, 13], led to insufficiently accurate prediction. The selection of a second, rotational compliance could improve the prediction of the second resonance peak significantly. This indicates, that it is recommendable to evaluate the probable modes of vibration of the assembly and subsequently select one or more coupling DOFs where these modes are observable.
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