54 P. Chavan et al. be observed in the backwards difference case. The estimation from beam decoupling was unsuccessful. It is thought that the compliance of the cylindrical beam is many magnitudes higher than the table and this influence cannot be effectively decoupled. For this reason, the compliances obtained from beam decoupling will not be used for the subsequent coupling calculations. 5.5 Variations of Coupling Calculations and Corresponding Results Based on the available compliance measurements, measurement techniques and the consideration of different rotational axis, several variations of the coupling matrices and calculations can be realized. The analysis of these combinations and their corresponding results are presented in this section. For the first analysis, let us consider and expand the term (BGBT) from Eq. (5.10), BGBT = Gc w +Gc t = ⎡ ⎢⎢ ⎣ ⎡ ⎢⎢ ⎣ G11 w G12 w G13 w G14 w G21 w G22 w G23 w G24 w G31 w G32 w G33 w G34 w G41 w G42 w G43 w G44 w ⎤ ⎥⎥ ⎦ + ⎡ ⎢⎢ ⎣ G11 t G12 t G13 t G14 t G21 t G22 t G23 t G24 t G31 t G32 t G33 t G34 t G41 t G42 t G43 t G44 t ⎤ ⎥⎥ ⎦ ⎤ ⎥⎥ ⎦ . (5.11) Here the termGc w corresponds to the compliance matrix at the four coupling points obtained from the reduced FE model of the workpiece. Gc t refers to the corresponding matrix for the coupling points of the table obtained through measurements. If two DOFs are considered for coupling, each term in the above matrices consists of, Gij = xi Fj xi Mj θi Fj θi Mj = Hij Lij Nij Pij . (5.12) Now, in the case of the workpiece compliances, every compliance in the term Gc w can be easily obtained from the simulation model. The same is not true for the measured compliance matrix Gc t of the table. In this case, the driving point translational FRFs in the z direction as well as the translational cross compliances in this direction are measurable with conventional measurement systems. Additionally, the direct rotational compliances at each coupling point can also be effectively estimated using the approaches shown in Sect. 5.4.2. However, the cross rotational and cross rotationaltranslational FRFs cannot be obtained. If the unascertainable compliances are replaced by ‘0’, the matrix Gc t can be represented as, G ij t = ⎧ ⎪⎪⎨ ⎪⎪⎩ for i =j, Hii t Lii t Nii t P ii t else H ij t 0 0 0 ⎫ ⎪⎪⎬ ⎪⎪⎭ . (5.13) Regarding the compliance matrix of the workpiece, there exist two options. The unascertainable DOFs of the table could either also be considered rigid for the workpiece or a fully occupied compliance matrix could be considered. In matrix form, CaseA, G ij w = ⎧ ⎪⎪⎨ ⎪⎪⎩ for i =j, Hii w Lii w Nii w P ii w else H ij t 0 0 0 ⎫ ⎪⎪⎬ ⎪⎪⎭ and Case B, G ij w = H ij w L ij w N ij w P ij w ∀ i,j. Subsequently, coupling equations were solved for both the cases for DOFs z and θx (obtained from T-Block) and the results of the prediction at DOF 5z along with the reference FRF are illustrated in Fig. 5.6. The comparison shows clearly that coupling only the measurable DOFs and considering others as rigid (Case A) led to a poor prediction. On the other hand, coupling with full workpiece compliance matrix results in a comparably better correspondence until about 150 Hz. Further analysis will show that still better results are achievable (Fig. 5.7).
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