10 Using SEMM to Identify the Joint Dynamics in Multiple Degrees of Freedom Without Measuring Interfaces 91 With Eq. (10.15) a coupled model can be created by weakening the compatibility condition. When including the joint in this manner, as compared to when including the joint as its own substructure, some extra assumptions are made: – The joint is mass-less. This assumption is based on the fact that the equilibrium condition introduced in Eq. (10.6) remains unaltered. Therefore, the forces acting on the boundary DoF of component AandBare still equal but oppositeat all times. Since this is no longer the case if a mass exists between the DoF (as this introduces counter-acting inertial forces), the joint is required to be mass-less. Alternatively, mass could be included beforehand by coupling (parts of) the joint-mass to either side of components Aand Busing standard LM-FBS. – The joint model YJ M×M is constructed such that, when the boundary forces act on it, the response u is the difference in response between the Mboundary DoF on component Aand B. There is therefore no information pertaining to the relation between DoF on any one side, i.e. it is assumed that there exists no coupling between the DoF on one interface which might not always be the case. These assumptions are recognized as similar to those found in the joint identification first introduced as inversesubstructuring, explained in detail in [7, 8]. These assumptions may, on first glance, limit the scope of joints that one might be interested in. However, both assumptions are valid for lightweight and small joint (such as friction contacts, glued contacts, welds, or even bolts when applied to a large structure). In fact, the linearity assumption accompanied by frequency based methods (small displacements and rotations) may already exclude the effects of these cross-coupling terms since they are bound to be non-linear. 10.2.1.2 Decoupling the Components to Obtain the Joint LM-FBS can be used to decouple components from full-systems as easily as it can be used to couple components to systems. It can be shown that decoupling is as easy as adding a negative model [5]. In this case specifically, the interest is in obtaining the joint dynamics from the systemYAJB, therefore decoupling is done by simply reversing Eq. (10.15). Figure 10.3 shows the process, which is indeed the reverse of what is shown in Fig. 10.2. It can be shown that standard decoupling is analogous to this equation inversion. This is done by first pre- and post-multiplying Eq. (10.15) by B and BT respectively and then solving it for YJ to obtain (10.16). YJ =−BYBT −BYBT B YBT −1 BYBT (10.16) where Yis the difference in FRF between the uncoupled and coupled model: Y=YAJB −Y (10.17) 10.2.2 System Equivalent Model Mixing It is shown in the previous sections that, with LM-FBS, components can be coupled with and without joints, and that a joint can be extracted from the system model. However, this still requires that the boundary DoF (c.f. the red markers in Fig. 10.3) are known. Measuring these DoF, especially in several directions and including rotational DoF, is often practically infeasible in the assembled state. So what if, alternatively, one can expand a measurable DoF-set to include boundary DoF? In this section an expansion method known as SEMM will be used to expand the measured internal DoF (c.f. the black markers in Fig. 10.3) to the required boundary DoF. YA YB YJ YAJB Fig. 10.3 The components YA and YB are decoupled from the full model YAJB. What is left is the dynamics of the joint represented byYJ
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