82 A. El Mahmoudi et al. The Moore-Penrose pseudoinverse of Rcan be used to determine q in the sense of the least squares, which minimizes the norm on residuals and forces R T μ. A comparable relation can be set up for nf > m forces f and virtual point forces/moments m. Thereby, one obtains: u=Ruq → q =(Ru)+u (7.5) m=R T ff → f = R T f +m (7.6) The inverted IDM matrices (Ru)+and R T f +are also known as transformation matrices. Thus, the measured FRFs Yuf can be transformed into a virtual point FRFYqm: q =Yqmm with Yqm (Ru)+Yuf R T f + (7.7) 7.2.3 Procedure of the iTPA In accordance with the proposal for standardization in [9] for the application of the method, the following five steps will be carried out: • FRF measurement (including virtual point transformation) • Operational measurement under various conditions • Blocked force calculation • On-board validation • Cross validation Blocked Force Calculation With respect to the in-situ blocked force TPA method, however, no transformation of the displacement is necessary, only of the forces. Thereby, Eq. (7.1) can be converted to the VPT for the force identification as follows: m eq 2 = Y assembly u4m2 +u op 4 with Y assembly u4m2 Y assembly u4f2 R T f2 + (7.8) The acceleration DoFs remain the same, while a characterization of forces and moments is created in the virtual point, allowing for the transfer of vibration through the interface. On-Board Validation The reliability of the identified force is now checked by using the on-board validation. As mentioned in the previous subsection, the target admittances Y assembly 32 and target sensor responses u op 3 are required. If a matrix-vector multiplication is performed with the transformed target admittance Y assembly u3m2 and the identified blocked force, the predicted system response u TPA 3 is obtained at the target point: u TPA 3 =Y assembly u3m2 m eq 2 ∼=u op 3 with Y assembly u3m2 Y assembly u3f2 R T f2 + (7.9) Cross Validation The cross validation verifies the actual goal of the in-situ TPA, which is to take the identified blocked forces from one assembly (here: soft configuration, see Fig. 7.3) to predict the response in another assembly (here: stiff structure). The
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