High-frequency Dynamic Characterization of Rubber Mounts through an Enhanced Virtual Point Transformation 159 A mere substitution of the projection matrices Twith T∗ in equation 4 would not yield the desired FRF matrix Hqm∈ R12×12. Instead, the projected FRF matrix would contain 12+nη rows and columns, due to the additional flexible DOF. Aiming to preserve only 6 DOF per virtual point, the flexible contribution to the response vector q ∈ R6×1 is evaluated at the virtual point and added to the rigid contribution ρ, cfr. equation 12. Regarding the excitation, equation 12 states that the excitation vector m∈R6×1, while acting on the virtual point, excites both rigid and flexible modes of the fixture. In relation to the measured FRF matrix, this addition boils down to a subsequent projection that maps the contribution of the (measured and excited) flexible modes onto the preserved 6 DOF of each virtual point, as in equation 13. qk =ρk +ϕq,kηk, gρ,k gη,k = mk ϕT q,kmk (12) H∗qm= I ϕq,1 0 0 0 0 I ϕq,2 T∗uHuf T∗f T I ϕq,1 0 0 0 0 I ϕq,2 T (13) The here obtained FRF matrix H∗qm describes how the excitations mk and (displacement-level) responses qk, as introduced in equation 1, relate to one another in the test assembly. It is thus fully equivalent to Hqm as obtained through equation 4. Equation 5 thus remains valid, i.e. one must merely substitute Hqmwith H∗qmto obtain the transfer dynamics of the mount itself. In conclusion, the proposed variant to the VPT is not an extended VPT, in the sense that the projected dynamics are captured using 6 DOF on each side of the interface, like is the case for the original VPT. It is enhanced, because in contrast to the original VPT, the flexibility of the fixtures is taken into account. Finally, one may remark – from the block-diagonal structure of all projection matrices in equation 13 – that the amount of flexible DOFnη may differ from side to side. One may thus apply the here proposed approach on side1and the original VPT on side 2. To maintain the generally recommended over-determination factor of 2, one should add two response channels and two excitations per flexible DOF considered. Similarly to the original VPT, the instrumentation and excitation layout can be checked based on the condition numbers of the matrices R∗. The following section presents a numerical validation, highlighting the benefits of extending the reduction basis with flexible eigenmodes. The experimental validation of the subsequent section demonstrates the added value of the enhanced VPT, which includes the second projection onto 6 DOF, for dynamic mount stiffness identification. Numerical Validation of the Enhanced VPT Figure 2 illustrates the numerical validation setup. It represents a 30 mm thick plate, modeled using linear hexahedral elements of size 5 mm. This 3D model was used previously for validating a polynomial fitting approach, assuming an idealized thin-plate strain field. The here presented validation includes a comparison against that polynomial fitting approach [12]. We consider only the projection of 8×3=24 response DOF (u1) onto the 6 DOF (q1) of 1 virtual point, for a single excitation DOFfh, cfr. equation 14. For this single-sided projection, equation 13 simplifies into 14. This simplification does not affect the validation, because dealing with a perfectly reciprocal model, one may interchange the notions of response DOF and reference DOF. H∗q1fh = I ϕq,1 T∗u,1Hu1fh = I ϕq,1 Ru,1 ϕu,1 †Hu1fh = I ϕq,1 Hρ1fh Hη1fh (14) The focus of this numerical validation is on the choice of extending Ru with flexible eigenmodes ϕu. Figure 3 shows that with only a single extra DOF, being the participation factor to one flexible eigenmode, a significantly higher sensor consistency is obtained than with three extra DOF in the case of the polynomial plate model fit. It must be noted that in this case, the single extra DOF is chosen to correspond to the second bending mode. This choice is made because, as one can see in Figure 2, the selected responses are located such that the first bending mode introduces little deformation. Even so, the benefits of using flexible eigenmodes are clear: with an equal amount of flexible DOF (nη =3), the reduction basis extended with eigenmodes clearly outperforms the one based on the polynomial plate model. Using mperfectly identified mode shapes, the consistency is virtually perfect up to the m-th eigenfrequency. Noting that in Figure 3, eigenfrequencies f1 to f12 are marked by vertical lines, this is illustrated also for m=12. In case of closely spaced modes (around 900 Hz), this holds approximately. Another significant benefit in practical applications, is the fact that one can freely select the amount of relevant DOF to consider. In Figure 3, for example, it is clear that by accounting for only two bending modes (nη = 2), one can already capture very accurately the local flexibility observed in the region of indicator points up to about 1500 Hz. In this case,
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