50 P. Chavan et al. The compatibility condition and force equilibrium from Eqs. (5.2) to (5.3) can be expressed as, Bu=0 and LTg =0 (5.7) where, Band LT are signed Boolean matrices for isolating the coupling DOFs in u= ur wuc t T and for localizing the set of unique DOFs from the set of merged or global DOFs respectively [15]. Furthermore, interface constraint force vector can be represented as, g =−BTλ (5.8) for obtaining a dual assembly of the structures. Where λ represent Lagrange multipliers, signifying the connection forces [16]. Substituting Eqs. (5.7) and (5.8) in Eq. (5.4), we get, ⎡ ⎣ Zw 0 BT w 0 Zc t BT t Bw Bt 0 ⎤ ⎦ ⎧ ⎨ ⎩ ur w uc t λ ⎫ ⎬ ⎭ = ⎧ ⎨ ⎩ fw fc t 0 ⎫ ⎬ ⎭ . (5.9) Eliminating λfrom the above equation and reformulating by replacing dynamic stiffness matrix Z=diag Zw, Zc t with dynamic compliance matrixZ−1 = G=diag Gw, Gc t gives [15], Gw+t =G−GBT BGBT −1 BG (5.10) where, Gw+t =dynamic compliance matrix of the assembled workpiece and table. As explained in the previous sub-section, the FRF matrix of the unconstrained workpiece Gw can be obtained from the FE model. This matrix contains direct and indirect compliances for all nodes at all degrees of freedom. For example, a workpiece model with three nodes (one internal and two interface nodes) with six DOFs each (x, y, z, θx, θy, θz) has a compliance matrix Gw of size (18 ×18) with 324 FRFs. In this case, the measured compliance matrix of the table Gc t with two interface nodes and six DOFs each, has a size of (12 ×12) with 144 FRFs. The measurement of such a matrix is not practical. In order to reduce the measurement and calculation work, it is necessary to first identify which coupling DOFs should be considered and which can be ignored for predicting the dynamic behavior at the target DOF. This is the subject of the next section. 5.3 Sensitivity Analysis The question of selection of appropriate coupling DOFs does not arise in case of simple two-dimensional structures like overhanging beams. Here the consideration of a translational compliance and a rotational compliance at the free end are enough for an exact coupling with another beam. However, in case of complex structures with three-dimensional mode shapes and multiple connection points, compliances in multiple directions have to be considered and the question of selection of coupling DOFs becomes relevant. In this section, the above problem is analyzed in a purely virtual environment. For this, a CAD model of the table of a 4-axis machine tool is created along with that of a real workpiece (Fig. 5.2, left). The machine structure under the table is not considered in order to reduce the complexity of vibration modes. The table is fixed to the ground through four spring-damper elements. The workpiece has material properties of C45 Steel and has two slots along the x-direction. One slot on the side facing the table along the complete length of the workpiece for increasing compliance. The second slot, on the side facing away from the table, is only until half of the workpiece length. This point ‘5’ can be expected to have the maximum compliance during the slot milling operation as it exactly between the points of attachment (1–4). The bolts are considered rigid for simplicity. The assembly of the two structures at the shown position on the table is treated as the reference structure and the translational FRF in z-direction at point 5 (5z5z) is the reference FRF. With the help of an FE model of the assembly, the reference FRF can be obtained along with the vibration modes (Fig. 5.2, right). There are two significant modes of vibration observable in the reference FRF. The first mode at 44 Hz represents a tilting rigid body mode about the x-axis (θx) due to the compliance of the spring-damper elements. The second mode at 2250 Hz corresponds to a flexible bending mode of mainly the workpiece about the y-axis (θy).
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