10 Using SEMM to Identify the Joint Dynamics in Multiple Degrees of Freedom Without Measuring Interfaces 93 Finally, SEMM requires the removed model which is none other than (a reduced form of) the parent model (c.f. Fig. 10.4). One can choose to formulate the removed model as the parent model itself, or as a reduced form of the parent model which contain only the internal DoF iA, iB. 2 An explanation on the differences is omitted in this paper, but is given in [4]. In this application, it is chosen for the removed and parent model to be the same size and thus the same: Yrem =Ypar (10.21) To solve (10.18) the compatibility and equilibrium constraints are applied as in LM-FBS. Compatibility requires: upar g −urem g =0 (10.22a) urem i −u ov i =0 (10.22b) The equilibrium condition reads: gpar b +g rem b =0 (10.23a) gpar i +g rem i +g ov i =0 (10.23b) Like before, the compatibility and equilibrium condition can be written in matrix-notation with the signed Boolean MatrixB: Compatibility : Bu =0, Equilibrium: g =BTλ, with B= upar i u par b urem i urem b uov i ⎡ ⎣ I 0 −I 0 0 0 I 0 −I 0 0 0 I 0 −I ⎤ ⎦ The formulation exactly follows the LM-FBS notation described above, thus it is solved with Eq. (10.10). It can be shown that from this solution the single-line equation of SEMM can be obtained. The derivation is omitted in this paper, but can also be found in [4]: YSEMM=Ypar gg −Y par gg Y par ig + Y par ii −Yov ii Y par gi +Ypar gg (10.24) Here we made use of the relation (10.21) to substitute the terms of the removed model. The SEMM model now contains the dynamics from the overlay model while including the boundary DoF’s required to extract the joint model with Eq. (10.16). When expanding a DoF-set, new information is extrapolated based on the information contained in the measurements of the overlay model. Unfortunately, such an extrapolation is generally erroneous. Consequently, any joint model extracted from the expanded SEMM model is also erroneous (although presumably less so than in the original parent model). If one assumes that the overlay model contains the correct dynamics, then the only error made is this extrapolation error. It comes solely from the parent model’s manifold (i.e. its modal content) differing from that of the correct overlay model. In other words, even though the SEMM method will alter the dynamics of the parent model to fit that of the overlay model, it can only do so based on the allowable modal directions of said parent model. This is because any deflection shape created by the parent model must still be a linear combination of its modeshapes. Figure 10.5 illustrates this simple fact with a short example of a clamped-free beam which—for simplicity—has only the first two modes. These are depicted in (a). The first mode-shape is relatively the same for both parent (red) and overlay (blue) model, yet the second differs. The normalized deflection shapes of the parent (red), overlay (blue) and resulting SEMM (green) model for a given actuation are illustrated in (b). When looking at the SEMM model, even though the shared DoF match as per design, the internal DoF cannot since the required deflection shape is not a linear combination of the parent mode-shapes. This is also the reason that no rigid connection should be used to couple the component models YA and YB to create the parent model YAB. Since, if that were the case, the compatibility condition would ensure that bA =bB in the entire modal 2Note that reduction in admittance space is done by simply removing the DoF from the matrix.
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